maximally edge-connected and vertex-connected graphs and ... · degree d(v)= dg(v) of a vertex v is...

Discrete Mathematics 308 (2008) 3265–3296www.elsevier.com/locate/disc

Maximally edge-connected and vertex-connected graphs anddigraphs: A survey

Angelika Hellwiga, Lutz Volkmannb,∗aInstitut für Medizinische Statistik, Universitätsklinikum der RWTH Aachen, PauwelsstraYe 30, 52074 Aachen, Germany

bLehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

Received 8 December 2005; received in revised form 15 June 2007; accepted 24 June 2007Available online 24 August 2007

Abstract

Let D be a graph or a digraph. If �(D) is the minimum degree, �(D) the edge-connectivity and �(D) the vertex-connectivity, then�(D)��(D)��(D) is a well-known basic relationship between these parameters. The graph or digraph D is called maximally edge-connected if �(D) = �(D) and maximally vertex-connected if �(D) = �(D). In this survey we mainly present sufficient conditionsfor graphs and digraphs to be maximally edge-connected as well as maximally vertex-connected. We also discuss the concept ofconditional or restricted edge-connectivity and vertex-connectivity, respectively.© 2007 Elsevier B.V. All rights reserved.

Keywords: Connectivity; Edge-connectivity; Vertex-connectivity; Restricted connectivity; Conditional connectivity; Super-edge-connectivity;Local-edge-connectivity; Minimum degree; Degree sequence; Diameter; Girth; Clique number; Line graph

1. Introduction

Numerous networks as, for example, transport networks, road networks, electrical networks, telecommunicationsystems or networks of servers can be modeled by a graph or a digraph. Many attempts have been made to determinehow well such a network is ‘connected’ or stated differently how much effort is required to break down communicationin the system between some vertices. Clearly, it is desirable that a network stays connected as long as possible in case offaults should arise. Two classical measures that indicate how reliable a graph or a digraph D is are the edge-connectivity�(D) and the vertex-connectivity or simply the connectivity �(D) of D. If �(D) is the minimum degree, then it is wellknown that �(D)��(D) and �(D)��(D). Thus, in order to construct reliable and fault-tolerant networks, sufficientconditions for graphs or digraphs D satisfying �(D)=�(D) (so-called maximally edge-connected) or �(D)=�(D) (so-called maximally connected) are of great interest. For the case that graphs or digraphs are maximally edge-connectedor connected also further connectivity parameters are needed to investigate the fault tolerance, such as super-edge-connectivity, restricted (edge-)connectivity or local-edge-connectivity.

Our objectives are to present results on (edge-)connectivity and maximally (edge)-connected graphs and digraphs.Furthermore, we take a look at the concept of conditional or restricted (edge-)connectivity.

∗ Corresponding author.E-mail addresses: [email protected] (A. Hellwig), [email protected] (L. Volkmann).

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3266 A. Hellwig, L. Volkmann / Discrete Mathematics 308 (2008) 3265–3296

For more information on connectivity in graphs we refer the reader to survey articles with different emphases byFàbrega and Fiol [47], Mader [103], Oellermann [113] and Xu [177] and to the Ph.D. thesis by Hellwig [69].

2. Some terminology and basic results

We consider finite (di-)graphs without loops and multiple edges. For any (di-)graph D the vertex set is denoted byV (D) and the edge set (sometimes called oriented edge set or arc set in the case that D is a digraph) by E(D). Wedefine the order of D by n = n(D) = |V (D)| and the size by m = m(D) = |E(D)|. If G is a graph, then the vertexdegree d(v) = dG(v) of a vertex v is the number of vertices of G adjacent to v. The minimum degree is denoted by� = �(G) and the maximum degree by � = �(G). The degree sequence of a graph G is defined as the nonincreasingsequence of the degrees of the vertices of G. For a vertex v ∈ V (G), the open neighborhood N(v) = NG(v) is the setof all vertices adjacent to v, and N [v] = NG[v] = N(v) ∪ {v} is the closed neighborhood of v. If e = uv is an edge ofa graph G, then let N(e) = NG(e) = (N(u) ∪ N(v)) − {u, v}.

For X ⊆ V (G) let G[X] be the subgraph of G induced by X. For a connected graph G, the distance d(u, v)=dG(u, v)

between two vertices u and v is the length of a shortest path from u to v. The diameter dm(G) of a connected graph Gis defined by dm(G) = maxu,v∈V (G) dG(u, v). If X and Y are two subsets of V (G), then we denote by (X, Y ) the set ofedges with one endpoint in X and the other one in Y and by d(X, Y ) = dG(X, Y ) = minx∈X, y∈Y dG(x, y) the distancebetween X and Y. The girth g of a graph is the length of its shortest cycle. A (�, g)-cage is a �-regular graph with ��2and girth g�3 having the least possible number of vertices. Let p be a positive integer. A (di-)graph D is said to bep-partite if it is possible to partition V (D) into p subsets V1, V2, . . . , Vp (called partite sets) such that every edge joinsa vertex of Vi to a vertex of Vj for i �= j . If G is a p-partite graph such that uv ∈ E(G) for every vertex u ∈ Vi andevery v ∈ Vj for i �= j , then we speak of a complete p-partite or complete multipartite graph. For p = 2, such graphsare called bipartite graphs. The complete bipartite graph Kp,q has the bipartition A, B with |A| = p and |B| = q suchthat for every pair of vertices u ∈ A and v ∈ B there exists the edge uv.

A vertex-cut in a graph G is a set X of vertices of G such that G − X is disconnected. The vertex-connectivity orsimply the connectivity � = �(G) of a graph G is the minimum cardinality of a vertex-cut of G if G is not complete,and �(G) = n − 1 if G is the complete graph Kn of order n. An edge-cut in a graph G is a set S of edges of G such thatG − S is disconnected. If S is a minimal edge-cut of a connected graph G, then, necessarily, G − S consists of exactlytwo components. Every nontrivial graph has an edge-cut. The edge-connectivity �=�(G) of a graph G is the minimumcardinality of an edge-cut of G if G is nontrivial and �(K1) = 0. We denote by G the complementary graph of a graphG. A graph G is self-complementary if G = G.

For other graph theory terminology we follow Bondy and Murty [21] or Chartrand and Lesniak [29].Connectivity, edge-connectivity and minimum degree of a graph are related by a basic inequality due to Whitney

[171] in 1932.

Theorem 2.1 (Whitney [171] 1932). For any graph G,

�(G)��(G)��(G). (1)

Chartrand and Harary [27] showed that this result is best possible in the sense that if a, b and c are positive integerssuch that a�b�c, then there exists a graph G with �(G) = a, �(G) = b and �(G) = c.

Because of (1), it is reasonable to say that a graph G is maximally connected if �(G) = �(G) and maximallyedge-connected if �(G) = �(G).

The following result on the diameter of a graph and its complement will be used later, and can be found in the bookby Bondy and Murty [21] on p. 14 as an exercise (for a proof cf. [148, p. 19]).

Theorem 2.2 (Bondy and Murty [21] 1976). If G is a graph of diameter dm(G)�4, then dm(G)�2.

Note that Theorem 2.2 generalizes the earlier result by Ringel [133] that a nontrivial self-complementary graph hasdiameter 2 or 3 as well as the observation that dm(G)�3 implies dm(G)�3 by Harary and Robinson [68].

A. Hellwig, L. Volkmann / Discrete Mathematics 308 (2008) 3265–3296 3267

3. Maximally edge-connected graphs

Sufficient conditions for equality of edge-connectivity and minimum degree for graphs were given by several authors.The starting point was an article by Chartrand [26] in 1966. He observed that if � is large enough, then the secondinequality in (1) becomes an equality.

Theorem 3.1 (Chartrand [26] 1966). If G is a graph of order n(G)�2�(G) + 1, then �(G) = �(G).

In 1974, Lesniak [91] proved the following strengthening of this result.

Theorem 3.2 (Lesniak [91] 1974). If G is a graph with

dG(u) + dG(v)�n(G) − 1

for all distinct non-adjacent vertices u and v, then �(G) = �(G).

One year later, Plesník [127] presented a nice condition based on the diameter that implies Theorems 3.1 and 3.2.

Theorem 3.3 (Plesník [127] 1975). If G is a graph of diameter dm(G)�2, then �(G) = �(G).

In 1989, Plesník and Znám [128] could relax the condition in Theorem 3.3 in the sense that some distances can begreater than 2.

Theorem 3.4 (Plesník and Znám [128] 1989). If in a connected graph G no four vertices u1, v1, u2, v2 with

d(u1, u2), d(u1, v2), d(v1, u2), d(v1, v2)�3

exist, then �(G) = �(G).

A pair of sets X, Y ⊂ V (G) with dG(X, Y ) = k, where k is a positive integer, is called k-distance maximal, if thereexist no sets X1 ⊇ X and Y1 ⊇ Y with X1 �= X or Y1 �= Y such that dG(X1, Y1) = k. With the concept of distancemaximal sets, we were able to generalize all the aforementioned results in this section.

Theorem 3.5 (Dankelmann and Volkmann [34] 1995). Let G be a connected graph. If for all 3-distance maximal pairsof sets X, Y ⊂ V (G) the condition �(G[X ∪ Y ]) = 0 is fulfilled, then �(G) = �(G).

To see that Theorem 3.4 is contained in Theorem 3.5, assume that X, Y ⊂ V (G) is a pair of 3-distance maximalsets. Then the hypothesis of Theorem 3.4 yields min{|X|, |Y |}�1, and the desired results follow immediately byTheorem 3.5.

Remark 3.6 (Dankelmann and Volkmann [34] 1995). Obviously, Theorems 3.1–3.4 only work for graphs with diam-eter at most 4. Fig. 1 shows a graph with � = 2 and arbitrary large diameter for which Theorem 3.5 guarantees equalityof minimum degree and edge-connectivity.

Fig. 1.


Corresponding examples exist also for every � = ��3. Now we present a further extension of Chartrand’sTheorem 3.1.

Theorem 3.7 (Hellwig and Volkmann [76] 2005). Let G be a connected graph. If for each edge e there exists at leastone vertex v incident with e such that d(v)�n(G)/2, then �(G) = �(G).

Goldsmith and White [63] have shown that in order to obtain the conclusion of Chartrand’s Theorem 3.1, one neednot require that each vertex have degree at least n/2, but only that the degrees be essentially ‘balanced’.

Theorem 3.8 (Goldsmith and White [63] 1978). Let G be a graph of order n. If the vertex set of G can be partitionedinto n/2 pairs of vertices ui and vi such that

dG(ui) + dG(vi)�n

for i = 1, 2, . . . , n/2, then �(G) = �(G).

We note that Theorem 3.8 implies Chartrand’s result only when n is even. The following degree sequence conditionof Bollobás includes Theorem 3.8 by Goldsmith and White for odd n.

Theorem 3.9 (Bollobás [19] 1979). Let G be a graph of order n�3 with degree sequence d1 �d2 � · · · �dn = �. If

k∑i=1

(di + dn−i )�kn − 1

for each integer k with 1�k� min{n/2 − 1, �}, then �(G) = �(G).

This sufficient condition of Bollobás [19] was generalized by Dankelmann and Volkmann [35] in 1997.

Theorem 3.10 (Dankelmann and Volkmann [35] 1997). Let G be a graph of order n�2 with degree sequenced1 �d2 � · · · �dn = �. If ��n/2 or if ��n/2 − 1 and

k∑i=1

(di + dn+i−�−1)�k(n − 2) + 2� − 1

for some integer k with 1�k��, then �(G) = �(G).

Hellwig and Volkmann [72] have given a degree sequence condition that only considers the lower end of the degreesequence and that improves a corresponding bound in [35].

Theorem 3.11 (Hellwig and Volkmann [72] 2003). Let G be a graph of order n�2 with degree sequenced1 �d2 � · · · �dn = �. If ��n/2 or if ��n/2 − 1 and

2k∑i=1

dn+1−i � max{k(n − 1) − 1, (k − 1)n + 2� − 1}

for some integer k with 2�k��, then �(G) = �(G).

In 1979, Goldsmith and Entringer [62] showed that, if for each vertex u of minimum degree, the vertices in theneighborhood of u have sufficiently large degree sum, then the graph is maximally edge-connected. This result is bestpossible of its type and implies also Chartrand’s Theorem 3.1.


Theorem 3.12 (Goldsmith and Entringer [62] 1979). Let G be a connected graph of order n�2. If

∑x∈N(u)

d(x)�

⎧⎨⎩⌊n

2

⌋2 −⌊n

2

⌋for all even n and for odd n�15,⌊n

2

⌋2 − 7 for odd n�15

for each vertex u of minimum degree, then �(G) = �(G).

In 1978, Boesch and Chen [15] found a further small extension of Chartrand’s theorem.

Theorem 3.13 (Boesch and Chen [15] 1978). Let G be a graph of order n with degree sequence d1 �d2 � · · · �dn=�.If dn−1 �n/2, then �(G) = �(G).

Boesch and Chen [15], Goldsmith [60,61] and Sampathkumar [134] studied minimum edge-cuts S of connectedgraphs G such that G − S consists of i components for an arbitrary integer i�2. Using this more general concept ofedge-cuts, Goldsmith [60] received extensions of Theorems 3.3 and 3.13.

Soneoka et al. [140] established a condition for maximally edge-connected graphs in terms of diameter and girth g,and they show that their condition is best possible for an infinite number of values of � when g is 4 or g is odd.

Theorem 3.14 (Soneoka et al. [139,140] 1985, 1987). Let G be graph with girth g. If dm(G)�g − 1 when g is oddor dm(G)�g − 2 when g is even, then �(G) = �(G).

Since the girth g�3 for every graph, Theorem 3.14 immediately leads to Plesník’s Theorem 3.3. Refinements ofTheorem 3.14 can be found in the articles by Fiol and Fàbrega [43,54].

In 1985, Esfahanian [40] has given a sufficient condition for maximally edge-connected graphs, depending on theorder, maximum and minimum degree as well as on the diameter. Two years later, a slight refinement of Esfahanian’scondition, which is best possible at least for diameters 3 and 4, was presented by Soneoka et al. [140].

Theorem 3.15 (Soneoka et al. [140] 1987). Let G be a graph with maximum degree ��3 and minimum degree �,then �(G) = �(G) when

n(G) > (� − 1)(� − 1)dm(G)−1 + � − 3

� − 2+ � − 1.

Let B be a bipartite graph. In the case that the diameter dm(B)�3, a result of Plesník and Znám [128] (see Corollary6.6 below) yields �(B) = �(B). If otherwise dm(B)�4, then Theorem 2.2 implies dm(B)�2 and Plesník’s Theorem3.3 leads to �(B)= �(B). Especially, it follows that if B is bipartite graph, then �(B)= �(B) or �(B)= �(B). Recently,we have proved that this statement is valid for all graphs.

Theorem 3.16 (Hellwig and Volkmann [80]). If G is an arbitrary graph, then �(G) = �(G) or �(G) = �(G).

Corollary 3.17 (Hellwig and Volkmann [80]). If G is a self-complementary graph, then �(G) = �(G).

Let R(G) =∑v∈V (G)1/d(v) be the inverse degree of a graph G without isolated vertices. Using Jensen’s inequality

[83], we proved:

Theorem 3.18 (Dankelmann et al. [33]). If G is a connected graph of minimum degree � and order n�� + 3, then�(G) = �(G) when

R(G) < 2 + 2

�(� + 1)+ n − 2�

(n − � − 2)(n − � − 1).

In [33] the authors present an infinite class of examples, which show that the bound in Theorem 3.18 is best possible,and they give a corresponding result for triangle-free graphs.


Theorem 3.19 (Fiol [52] 1993). If G is a �-regular graph with n(G) > �dm(G)−1 + 2� − 2, then �(G) = �.

Theorem 3.20 (Wang et al. [162] 2003). Every (�, g)-cage with ��2 and odd girth g is maximally edge-connected.

Theorem 3.21 (Lin et al. [97] 2005). Every (�, g)-cage with ��2 and even girth g is maximally edge-connected.

Combining the last two results, we obtain the next theorem, which was found independently by Moriarty andChristopher [112].

Theorem 3.22 (Moriarty and Christopher [112] 2005). Every (�, g)-cage with ��2 is maximally edge-connected.

In addition, a theorem of Mader [102] says that every connected vertex-symmetric graph is maximally edge-connected.

4. Maximally edge-connected digraphs

Many of the results in Section 3 have analogues for digraphs, and we will see below that these analogues lead in anatural way immediately to the corresponding statements for graphs.

For a vertex v ∈ V (D) of a digraph D, the degree of v, denoted by d(v) = dD(v), is defined as the minimum valueof its out-degree d+(v) = d+

D(v) and its in-degree d−(v) = d−D(v). The minimum out-degree, minimum in-degree,

maximum out-degree and maximum in-degree of a digraph D are denoted by �+(D), �−(D), �+(D) and �−(D). Inaddition, let �(D) = min{�+(D), �−(D)} and �(D) = max{�+(D), �−(D)} be the minimum and maximum degreesof D, respectively. The degree sequence of D is the nonincreasing sequence of the degrees of the vertices of D. Forany vertex v of a digraph D, we denote the set of out-neighbors and in-neighbors of v by N+(v) = N+

D(v) andN−(v) = N−

D(v), respectively.If D is a digraph and X ⊆ V (D), then D[X] is the subdigraph induced by X, and its underlying graph G(D) is

the graph obtained by replacing each oriented edge of D by an undirected edge joining the same pair of vertices. Thecomplete digraph K∗

n is of order n and minimum degree n− 1. The complete bipartite digraph K∗p,q has the bipartition

A, B with |A| = p and |B| = q such that for every pair of vertices u ∈ A and v ∈ B there exist both (oriented) edgesuv and vu.

A digraph D is strongly connected or simply strong if for every pair u, v of vertices there exists a directed pathfrom u to v in D. A digraph with at least k + 1 vertices is k-connected, if for any set X of at most k − 1 vertices, thesubdigraph D − X is strong. The connectivity �(D) of a digraph D is defined as the largest value of k such that D isk-connected. A digraph D is k-edge-connected if for any set S of at most k − 1 edges the subdigraph D − S is strong.The edge-connectivity �(D) of a digraph D is defined as the largest value of k such that D is k-edge-connected.

In 1971, Geller and Harary have written in the introduction of their article [58]: ‘Connectivity of graphs has beenextensively investigated. On the other hand, connectivity in digraphs has until recently been almost completely neglected.In this article, we begin with an expository review of connectivity concepts and results concerning both graphs anddigraphs.’ Among other things, Geller and Harary received an analogue to Whitney’s Theorem 2.1 for digraphs.

Theorem 4.1 (Geller and Harary [58] 1971). If D is a digraph, then

�(D)��(D)��(D). (2)

In view of (2), we call a digraph D maximally connected if �(D)=�(D) and maximally edge-connected if �(D)=�(D).The associated digraph D(G) of a graph G is obtained by replacing each edge of G by a pair of two mutually

opposite oriented edges. The next observation is simple but important.

Observation 4.2. If G is a graph and D(G) its associated digraph, then there is a one–one correspondence betweenpaths in G and directed paths in D(G). This immediately leads to �(G) = �(D(G)) and �(G) = �(D(G)).

The distance d(u, v)=dD(u, v) from u to v in a digraph D is the length of a shortest directed path from u to v in D andthe distance d(X, Y )=dD(X, Y ) from a vertex set X to a vertex setY in D is given by dD(X, Y )=minx∈X,y∈Y dD(x, y).


The diameter dm(D) of a strong digraph D is defined by dm(D)= maxu,v∈V (G) dD(u, v). A pair of vertex sets X and Yof D with distance dD(X, Y ) = k, k ∈ N, is called k-distance maximal if there exist no vertex sets X1 ⊇ X and Y1 ⊇ Y

with X1 �= X or Y1 �= Y such that dD(X1, Y1) = k.

Theorem 4.3 (Hellwig and Volkmann [72] 2003). Let D be a strong digraph. If for all 3-distance maximal pairs ofvertex sets X and Y there exists an isolated vertex in D[X ∪ Y ], then �(D) = �(D).

Applying Observation 4.2, it is easy to see that Theorem 3.5 follows immediately from Theorem 4.3, and thusTheorem 4.3 includes also Theorems 3.1–3.4 as well as the digraph versions of these results (cf. [72]).

Corollary 4.4 (Geller and Harary [58] 1971, Jolivet [85] 1972). If D is a strong digraph, then �(D) = �(D) if any ofthe following conditions holds:

(i) n(D)�2�(D) + 1,(ii) d+

D(u) + d−D(v)�n(D) − 1 for all pairs of non-adjacent vertices u and v,

(iii) dm(D)�2,(iv) there exist no four vertices u1, v1, u2, v2 with

dD(u1, u2), dD(u1, v2), dD(v1, u2), dD(v1, v2)�3.

Notice that Plesník’s Theorem 3.3 is a direct consequence of the older result (iii) of Corollary 4.4 by Jolivet [85].For an oriented graph D, a digraph without directed cycles of length two, Ayoub and Frisch [1] have given a weaker

condition than that of Corollary 4.4(i).

Theorem 4.5 (Ayoub and Frisch [1] 1970). If D is an oriented graph with minimum degree �(D)�(n(G) + 2)/4,then �(D) = �(D).

Next we list the digraph versions of Theorems 3.7, 3.8, 3.10 and 3.11.

Theorem 4.6 (Hellwig and Volkmann [77] 2005). Let D be a strong digraph of order n. Then �(D)=�(D), if for eachedge e = uv the following two inequalities are valid:

max{d+(u), d+(v)}�⌊n

2

⌋and max{d−(u), d−(v)}�

⌊n

2

⌋.

Theorem 4.7 (Xu [175] 1994). Let D be a digraph of order n. If the vertex set of G can be partitioned into n/2 pairsof vertices ui and vi such that

dD(ui) + dD(vi)�n

for all i = 1, 2, . . . , n/2, then �(G) = �(G).

A short proof of Theorem 4.7 is due to Dankelmann and Volkmann [35]. This proof shows that one can relax thecondition on the degrees slightly by allowing one pair of vertices to have degree sum n − 1.

Theorem 4.8 (Dankelmann and Volkmann [35] 1997). Let D be a digraph of order n�2 with degree sequenced1 �d2 � · · · �dn = �. If ��n/2 or if ��n/2 − 1 and

k∑i=1

(di + dn+i−�−1)�k(n − 2) + 2� − 1

for some integer k with 1�k��, then �(D) = �(D).

For the special case of oriented graphs the following weaker condition implies � = �.


Theorem 4.9 (Volkmann [152] 2006). Let D be an oriented graph of order n�2 with degree sequence d1 �d2 � · · ·�dn = �. If ��(n − 1)/4� or if ��(n − 1)/4� − 1 and

k∑i=1

(di + dn+i−2�−1)�k(n − k − 1) + 2� − 1

for some integer k with 1�k�2� + 1, then �(D) = �(D).

Theorem 4.10 (Hellwig and Volkmann [72] 2003). Let D be a digraph of order n�2 with degree sequenced1 �d2 � · · · �dn = �. If ��n/2 or if ��n/2 − 1 and

2k∑i=1

dn+1−i � max{k(n − 1) − 1, (k − 1)n + 2� − 1}

for some integer k with 2�k��, then �(D) = �(D).

Hellwig and Volkmann [77] have also given a digraph version and different generalizations of Theorem 3.12 byGoldsmith and Entringer [62]. However, the statements of their results are intricate and long, and hence we refer theinterested reader to the original text [77].

In 1985, Imase et al. [81] presented a related result to Theorem 3.15 for digraphs.

Theorem 4.11 (Imase et al. [81] 1985). If D is a digraph with maximum degree ��2 and minimum degree �, then�(D) = �(D) when

n(D) > (� − 1)

(�dm(D)−1 − 1

� − 1+ � + 1

).

5. Maximally edge-connected (oriented) graphs with given clique number

We now turn our attention to graphs with given clique number. Recall that the clique number � = �(G) of a graphG is the maximum cardinality of a complete subgraph of G. A first such result was presented by Volkmann [147] forp-partite graphs.

Theorem 5.1 (Volkmann [147] 1989). If G is a p-partite graph with p�2, then �(G) = �(G) when

n(G)�2

⌊�(G)

p

p − 1

⌋− 1.

In the same article, Volkmann [147] pointed out that Theorem 5.1 remains valid for p-partite digraphs.If G is a graph with clique number �(G)�p, then the well-known theorem of Turán [143] leads to the fundamental

upper bound

2m(G)� p

p − 1n(G)2 (3)

for the size m(G) of the graph G. Using Turán’s inequality (3), some extensions and supplements of Theorem 5.1 weregiven.

Theorem 5.2 (Dankelmann and Volkmann [34] 1995). Let G be a graph with clique number �(G)�p. Ifn(G)�2�(G)p/(p − 1) − 1, then �(G) = �(G).

Volkmann [147] has constructed examples which achieve the bound in Theorem 5.1 and thus also this one in Theorem5.2. For oriented graphs one can relax the condition in Theorems 5.1 and 5.2.


Theorem 5.3 (Volkmann [155]). Let D be an oriented graph such that the clique number �(G(D))�p in its underlyinggraph G(D). Then �(D) = �(D) when

n(D)�4

⌊�(D)

p

p − 1

⌋− 1.

Using some results in [155], Volkmann [157] recently presented degree sequence conditions for maximally edge-connected and super-edge-connected oriented graphs depending on the clique number.

Next we consider conditions involving the degree sequence for graphs. We start with a generalization ofTheorem 5.2.

Theorem 5.4 (Dankelmann and Volkmann [36] 2000). Let G be a graph of order n, clique number �(G)�p and withdegree sequence d1 �d2 � · · · �dn = �. If n�2�p/(p − 1) − 1 or if n�2�p/(p − 1)�2p and

k∑i=1

di +(2p−1)k∑

i=1

dn+1−i �k(p − 1)n + 2� − 1

for some integer k with 1�k��/(p − 1), then �(G) = �(G).

Theorem 5.5 (Dankelmann and Volkmann [36] 2000). Let G be a graph of order n, clique number �(G)�p and withdegree sequence d1 �d2 � · · · �dn = �. If n�2�p/(p − 1) − 1 or if n�2�p/(p − 1)�2p and

2pk∑i=1

dn+1−i �k(p − 1)n + 2

⌊k(� − 1)

�/(p − 1)⌋

+ 1 (4)

for some integer k with 1�k��/(p − 1), then �(G) = �(G).

Theorem 5.5 is best possible in the sense that one cannot delete the last one in (4), and it implies an earlier resultby Dankelmann and Volkmann [35] on bipartite graphs. The next strengthening of Theorem 5.2 is independent ofTheorems 3.11, 3.12, 5.4 and 5.5.

Theorem 5.6 (Volkmann [149] 2003). Let G be a graph of order n�6, clique number �(G)�p and with degreesequence d1 �d2 � · · · �dn = �. Furthermore, let � = 1 when n is even and � = 0 when n is odd. If ��n/2 or if��n/2 − 1 and

�+1∑i=1

dn+1−i �(� + 1) · p − 1

p· n + 1 + �

2− 2� + 2

p(n − 3 + �),

then �(G) = �(G).

In the case that G is of even order, Volkmann [149] could relax the condition in Theorem 5.6 slightly.

Theorem 5.7 (Volkmann [149] 2003). Let G be a graph of order n�6, clique number �(G)�p and with degreesequence d1 �d2 � · · · �dn = �. If ��n/2 or if ��n/2 − 1 and

2�+2∑i=1

dn+1−i �(� + 1) · p − 1

p· (n + 2) − 4� + 4

p(n − 2),

then �(G) = �(G).

Examples in [149] show that Theorems 5.6 and 5.7 are best possible when n is odd and n is even, respectively.


6. Maximally edge-connected bipartite (di-)graphs

The special case p = 2 in Theorem 5.1 leads to a 1988 result by Volkmann [146] on maximally edge-connectedbipartite graphs.

Corollary 6.1 (Volkmann [146] 1988). If G is a bipartite graph of order n(G)�4�(G) − 1, then �(G) = �(G).

During the past two decades various improvements, extensions and supplements of Corollary 6.1 were obtained.In the sequel let D be a bipartite (di-)graph with bipartition V (D) = V ′ ∪ V ′′. We adopt the convention that, for

every subset X of V (D), we denote the set X ∩ V ′ by X′ and X ∩ V ′′ by X′′. A pair of vertex sets X and Y of a bipartitegraph or digraph D with dD(X′, Y ′) = k, and dD(X′′, Y ′′) = k, k ∈ N, is called (k, k)-distance maximal, if there existno vertex sets X1 ⊇ X and Y1 ⊇ Y with X1 �= X or Y1 �= Y such that dD(X′

1, Y′1) = dD(X′′

1 , Y ′′1 ) = k.

Theorem 6.2 (Hellwig and Volkmann [72] 2003). Let D be a strong connected bipartite digraph. If for all (4,4)-distance maximal pairs of vertex sets X and Y there exists an isolated vertex in D[X ∪ Y ], then �(D) = �(D).

Corollary 6.3 (Hellwig and Volkmann [72] 2003). Let D be a bipartite digraph with bipartition V (D) = V ′ ∪ V ′′. Ifd(x, y) = 2 for all different x, y ∈ V ′, then �(D) = �(D).

Corollary 6.4 (Dankelmann and Volkmann [34] 1995). Let G be a bipartite graph with bipartition V (G) = V ′ ∪ V ′′.If d(x, y) = 2 for all different x, y ∈ V ′, then �(G) = �(G).

Corollary 6.5 (Fàbrega and Fiol [45] 1996). If D is a bipartite digraph of diameter at most 3, then �(D) = �(D).

Corollary 6.6 (Plesník and Znám [128] 1989). If G is a bipartite graph of diameter at most 3, then �(G) = �(G).

Since by a fundamental 1916 result of König [89] a graph is bipartite if and only if it contains no cycle of odd length,the next condition by Fàbrega and Fiol [45] depending on the girth g implies Corollary 6.6 and hence Corollary 6.1.

Theorem 6.7 (Fàbrega and Fiol [45] 1996). If G is a bipartite graph of minimum degree ��2 and girth g such thatdm(G)�g − 1, then �(G) = �(G).

As a further generalization of Corollary 6.1, Balbuena and Carmona [2] have given in 2001 a degree condition forbipartite digraphs.

Theorem 6.8 (Balbuena and Carmona [2] 2001). If D is a bipartite digraph such that

d+D(u) + d−

D(v)�⌈

n(D) + 1

2

⌉

for all pairs of vertices u, v with dD(u, v)�4, then �(D) = �(D).

An immediate corollary, due to Dankelmann and Volkmann [34], now follows.

Corollary 6.9 (Dankelmann andVolkmann [34] 1995). If G is a bipartite graph of order n such that d(u)+d(v)��(n+1)/2� for all non-adjacent vertices u and v, then �(G) = �(G).

Corollary 6.9 is a consequence of a more general result by Dankelmann and Volkmann [34] on bipartite graphs. In2003, Hellwig and Volkmann [72] could prove the following digraph version of this result. For a non-complete digraphD let

NC2(D) = min{|N+(u) ∪ N+(v)|, |N−(u) ∪ N−(v)| : u, v ∈ V (D) , d(u, v) = 2}.


Theorem 6.10 (Hellwig and Volkmann [72] 2003). Let D be a strong connected bipartite digraph of order n�3. IfNC2(D)��(n + 1)/4�, then �(D) = �(D).

The following degree sequence condition is an analogue to Theorems 3.10 and 4.8 for bipartite (di-)graphs.

Theorem 6.11 (Dankelmann and Volkmann [35] 1997). Let D be a bipartite (di-)graph of order n with degree sequenced1 �d2 � · · · �dn = �. If ��(n + 1)/4 or if ��n/4 and

k∑i=1

(di + dn+1−2�+k−i )�k(n − 2�) + 2� − 1

for some integer k with 1�k�2�, then �(D) = �(D).

As an application of Theorem 6.11, we could prove two analogous results to Theorems 3.8 and 4.7 for bipartitedigraphs, where, for even n, the second theorem is slightly better than the first one.

Theorem 6.12 (Hellwig and Volkmann [72] 2003). Let D be a bipartite (di)-graph of order n�2 and minimum degree�. If there are n/2 disjoint pairs of vertices (vi, wi) with d(vi)+ d(wi)�n− 2�+ 1 for all i = 1, 2, . . . , n/2, then�(D) = �(D).

Theorem 6.13 (Hellwig and Volkmann [72] 2003). Let D be a bipartite (di)-graph of even order n�2 and minimumdegree �. If there are n/2−1 disjoint pairs of vertices (vi, wi) with d(vi)+d(wi)�n−2�+1 for all i=1, 2, . . . , n/2−1and one further pair (v, w) with d(v) + d(w)�n − 2�, then �(D) = �(D).

Examples show that Theorem 6.13 is best possible in the sense that the condition that there are n/2 − 2 disjoint pairsof vertices (vi, wi) with d(vi) + d(wi)�n − 2� + 1 for i = 1, 2, . . . , n/2 − 1 and two further pairs with degree sumexactly n − 2� does not guarantee � = �.

Recently, Volkmann [152] gave an analogue to Theorem 6.13 for oriented bipartite graphs.

Theorem 6.14 (Volkmann [152] 2006). Let D be an oriented bipartite graph of even order n and minimum degree �.If there are n/2 disjoint pairs of vertices (vi, wi) with

d(vi) + d(wi)�n + 1

2− 2�

for all i = 1, 2, . . . , n/2, then �(D) = �(D).

7. Super-edge-connectivity

Bauer et al. [11] proposed the concept of super-edge-connectedness. A (di-)graph D is called super-edge-connectedor super-� if every minimum edge-cut is trivial; that is, if every minimum edge-cut consists of edges incident with avertex of minimum degree. Thus every super-edge-connected graph is also maximally edge-connected. The study ofsuper-edge-connected (di-)graphs has a particular significance in the design of reliable networks (cf. [14]). Most of theknown sufficient conditions for a (di-)graph D to be super-� are closely related to those ones in the preceding sections.The first such condition was given by Kelmans [87] in 1972.

Theorem 7.1 (Kelmans [87] 1972). If G is a graph with �(G)�(n(G) + 1)/2, then G is super-�.

In connection with Corollary 4.4(iii) by Jolivet [85], Fiol [51] has shown that most of the digraphs with diameter 2are even super-�. Let 2K∗

� be any digraph obtained by joining two disjoint copies of K∗� by some edges in such a way

that the minimum degree of 2K∗� is �.


Theorem 7.2 (Fiol [51] 1992). Let D be a digraph of order n and minimum degree �. Then D is super-� if any of thefollowing conditions hold:

(i) dm(D) = 2 and D contains no K∗� with all its vertices of out-degree � or all its vertices of in-degree �,

(ii) d+D(u) + d−

D(v)�n for all pairs u, v of vertices in D such that d(u, v)�2, and D is different from 2K∗n/2,

(iii) d+D(u) + d−

D(v)�n + 1 for all pairs u, v of vertices in D such that d(u, v)�2.

Corollary 7.3 (Fiol [51] 1992, Lesniak [91] 1974). Let G be a graph of order n and minimum degree �. Then G issuper-� if any of the following conditions hold:

(i) dm(G) = 2 and G contains no K� with all its vertices of degree �,(ii) dG(u) + dG(v)�n for all pairs of non-adjacent vertices u and v and D is different from Kn/2 × K2,

(iii) dG(u) + dG(v)�n + 1 for all pairs of non-adjacent vertices u and v.

The next two results of Fiol [51] on oriented graphs are closely related to Theorem 4.5 by Ayoub and Frisch [1].

Theorem 7.4 (Fiol [51] 1992). An oriented graph D is super-� when

�+(D) + �−(D)�n(D)/2 + 1.

Corollary 7.5 (Fiol [51] 1992). An oriented graph D is super-� when

�(D)�n(D)/4 + 1.

Note that, according to Theorem 7.2(i), any oriented graph with diameter 2 is super-�. However, this result andTheorem 7.4 (or Corollary 7.5) do not imply each other.

In connection with Theorems 3.10, 3.11, 4.8 and 4.10, Volkmann [150] has given different degree sequence conditionsfor (di-)graphs to be super-�.

Theorem 7.6 (Volkmann [150] 2003). Let D be a (di-)graph of order n�2 with degree sequence d1 �d2 � · · · �dn=�.If ��n/2 + 1 or if ��n/2 and

k∑i=1

(di + dn+i−�)�k(n − 2) + 2� + 1

for some integer k with 1�k��, then D is super-�.

Theorem 7.7 (Volkmann [150] 2003). Let D be a (di-)graph of order n with degree sequence d1 �d2 � · · · �dn=��2.If ��n/2 + 1 or if ��n/2 and

2k∑i=1

dn+1−i � max{k(n − 1) + 1, (k − 1)n + 2� + 1}

for some integer k with 1�k��, then D is super-�.

The next three results are closely related to Theorems 3.14 and 3.15 for graphs as well as to Theorem 4.11 fordigraphs. Knowing the girth helps to ascertain sufficient conditions for graphs to be super-�.

Theorem 7.8 (Fàbrega and Fiol [43] 1989, Fiol et al. [55] 1990). Let G be a graph with girth g. If dm(G)�g − 2when g is odd or dm(G)�g − 3 when g is even, then G is super-�.

For refinements of Theorems 3.16 and 7.8, we refer the reader to the article [44] by Fàbrega and Fiol.


Theorem 7.9 (Soneoka [138] 1992). If G is a graph with maximum degree ��3 and minimum degree �, then G issuper-� when

n(G) > �

((� − 1)dm(G)−1 − 1

� − 2+ 1

)+ (� − 1)dm(G)−1.

Soneoka [138] has constructed regular graphs G, which show that the bound in Theorem 7.9 is best possible at leastfor diameters 2, 3, 4 and 6. A generalization of Theorem 7.9 can be found in a paper by Fiol [53].

Theorem 7.10 (Soneoka [138] 1992). Let D be a digraph with maximum degree ��2 and minimum degree �. ThenD is super-� when

n(D) > �

(�dm(D)−1 − 1

� − 1+ 1

)+ �dm(D)−1.

Examples by Soneoka [138] demonstrate that the given bound is best possible at least for diameters 2 and 3.Volkmann [151] has proved that graphs which fulfill the conditions of Theorem 5.2 are also super-� in the most

cases.

Theorem 7.11 (Volkmann [151] 2004). Let G be a graph of order n, clique number �(G)�p and minimum degree�. Then G is super-� when

n�2

⌊p�

p − 1

⌋− 3 and � = p or � = k(p − 1),

n�2

⌊p�

p − 1

⌋− 1 otherwise.

Theorem 7.12 (Fiol [51] 1992). Let D be a bipartite (di-)graph with minimum degree ��3. If d+D(u) + d−

D(v)�n(D)/2 + 2 for all pairs of vertices u, v with dD(u, v)�3, then D is super-�.

Corollary 7.13 (Fiol [51] 1992). Let D be a bipartite (di-)graph with minimum degree ��3. If ��(n(D) + 3)/4�,then D is super-�.

Theorem 3.4 in Fiol’s article [51] states that �+ + �− ��(n + 1)/4� is sufficient for an oriented bipartite graph to besuper-�. However, the next example will show that this is not valid in general.

Example 7.14 (Volkmann [155]). Let T be the oriented bipartite graph of order 14 with the partition sets X ={x1, x2, x3, x4, x

′1, x

′2, x

′3} and Y ={y1, y2, y3, y

′1, y

′2, y

′3, y

′4} such that y1 → x1 → y2 → x2 → y3 → x3 → y2 → x′

2,y3 → x4 → y2, x1 → y3, {x2, x3, x4} → y1 → x′

1, y′1 → x′

1 → y′2 → x′

2 → y′3 → x′

3 → y′2, x′

2 → y′4 → x′

3 → y′1,

x′1 → {y′

3, y′4}, x′

2 → y′1 and y′

i → xj for 1� i, j �4.Now n(T ) = 14, �(T ) = �+(T ) = �−(T ) = 2, 4 = �+(T ) + �−(T ) = �(n(T ) + 1)/4� and thus �(T ) = �(T ) = 2.

However, T is not super-�, since S = {y1x′1, y2x

′2} is a minimum edge-cut.

Corresponding examples also exist for every �+ =�− �3. But a slightly stronger condition than this one of Fiol [51]implies that an oriented bipartite graph is super-�.

Theorem 7.15 (Volkmann [155]). Let D be an oriented bipartite graph of order n. Then D is super-� when

�+(D) + �−(D)�⌈

n + 3

4

⌉.

In 2004, Hellwig and Volkmann [75] have shown that bipartite graphs of minimum degree ��3 and of order n withn ≡ 0 (mod 4) or n ≡ 1 (mod 4), which fulfill the conditions of Theorem 6.10, are even super-�.


Theorem 7.16 (Hellwig and Volkmann [75] 2004). If D is a strongly connected bipartite (di-)graph of order n andminimum degree ��3, then D is super-� when

NC2(D)�⌈

n + 3

4

⌉.

Theorem 7.16 is an improvement of Corollary 7.13. Examples in [75] show that this result is best possible. The cycleC4 of length four demonstrates that Theorem 7.16 is not valid for �= 2. The paper [75] by Hellwig and Volkmann alsocontains a generalization of the following result by Volkmann.

Theorem 7.17 (Volkmann [150] 2003). Let D be a bipartite (di-)graph of order n with degree sequence d1 �d2 � · · ·�dn = ��3. If ��(n + 3)/4� or if ��(n + 3)/4� − 1 and

k∑i=1

(di + dn+i−2�+1)�(k + 1)(n − 2�) + 2� + 1

for some integer k with 1�k�2� − 1, then D is super-�.

Marcote and Balbuena [104] have obtained an extension of Theorem 3.20 by Wang et al. [162] on cages with oddgirth.

Theorem 7.18 (Marcote and Balbuena [104] 2004). Every (�, g)-cage with ��3 and odd girth g is super-edge-connected.

Two years later, Lin et al. [96] proved the same for cages with even girth.

Theorem 7.19 (Lin et al. [96] 2006). Every (�, g)-cage with ��3 and even girth g is super-edge-connected.

Combining the last two results, we arrive at the following improvement of Theorem 3.22.

Corollary 7.20. All (�, g)-cages with ��3 are super-edge-connected.

Cycles Cn of length n�4 show that in Theorems 7.18 and 7.19 as well as in Corollary 7.20 the condition ��3 isnecessary.

More results on super-edge-connectivity can be found, for example, in the papers by Balbuena et al. [7], Boesch andTindell [16], Boesch and Wang [17], Boland and Ringeisen [18], Chen et al. [31], Fàbrega and Fiol [45], Fiol [50],Hamidoune and Tindell [66], Li and Li [94], Meng [108], Pelayo et al. [126], Shieh [137] and Volkmann [160].

8. Maximally local-edge-connected (di-)graphs

The local-edge-connectivity �(u, v) of two vertices u and v in a (di-)graph D is the maximum number of edge-disjointpaths from u to v in D. Clearly, �(u, v)� min{d+(u), d−(v)} for all pairs u and v of vertices in D (if D is a graph, thenlet d+(x) = d−(x) = d(x) for a vertex x in D). We call a (di-)graph D maximally local-edge-connected when

�(u, v) = min{d+(u), d−(v)}for all pairs u and v of vertices in D.

In 2000, Fricke et al. [56] have shown that some known sufficient conditions that guarantee �(G)= �(G) for a graphG also guarantee that G is maximally local-edge-connected. The next observation shows that the results of Fricke etal. [56] generalize the corresponding known one.

Observation 8.1. If a (di-)graph D is maximally local-edge-connected, then �(D) = �(D).


Proof. Since D is maximally local-edge-connected, we have �(u, v) = min{d+(u), d−(v)} for all pairs u and v ofvertices in D. Thus, it follows from the well-known theorem of Menger [111] that

�(D) = minu,v∈V (D)

{�(u, v)} = minu,v∈V (D)

{min{d+(u), d−(v)}} = �(D). �

Theorem 8.2 (Fricke et al. [56] 2000). If G is a graph with dm(G)�2, then �(u, v) = min{d(u), d(v)} for all pairsu and v of vertices in G.

Theorem 8.3 (Fricke et al. [56] 2000). Let p be an integer withp�2 and let G be a p-partite graph. Ifn(G)�2�(G)p/

(p − 1) − 1, then �(u, v) = min{d(u), d(v)} for all pairs u and v of vertices in G.

In view of Observation 8.1, Theorems 3.3 and 5.1 are immediate consequences of Theorems 8.2 and 8.3, respectively.Hellwig and Volkmann [73] have extended Theorem 8.2 to digraphs.

Theorem 8.4 (Hellwig and Volkmann [73] 2004). If D is a digraph with dm(D)�2, then �(u, v)=min{d+(u), d−(v)}for all pairs u and v of vertices in D.

Theorem 8.4 is also a refinement of Corollary 4.4(iii) by Jolivet [85]. Using again Turán’s inequality (3), we alsoobtained an improvement of Theorem 8.3.

Theorem 8.5 (Hellwig and Volkmann [73] 2004). Let p�2 be an integer and let G be a graph with clique number�(G)�p. If n(G)�2�(G)p/(p − 1) − 1, then �(u, v) = min{d(u), d(v)} for all pairs u and v of vertices in G.

An analogue of Theorem 8.3 for p-partite digraphs is also valid.

Theorem 8.6 (Hellwig and Volkmann [73] 2004). Let p be an integer with p�2 and let D be a p-partite digraph. Ifn(D)�2�(D)p/(p − 1) − 1, then �(u, v) = min{d+(u), d−(v)} for all pairs u and v of vertices in D.

As a strengthening of Corollary 6.9, we finally present a sharp sufficient condition for bipartite (di-)graphs to bemaximally local-edge-connected.

Theorem 8.7 (Hellwig and Volkmann [73] 2004). Let D be a bipartite (di-)graph of order n and minimum degree��2 with bipartition V ′ ∪ V ′′. If d(x) + d(y)�(n + 1)/2 for each pair of vertices x, y ∈ V ′ and each pair of verticesx, y ∈ V ′′, then �(u, v) = min{d+(u), d−(v)} for all pairs u and v of vertices in D.

As a generalization of Theorem 4.5 by Ayoub and Frisch [1], Volkmann [154] recently proved the following result.

Theorem 8.8 (Volkmann [154]). Let D be an oriented graph of order n and minimum degree �. If n�4� + 1, then Dis maximally local-edge-connected.

The next theorem shows that one can relax the condition in Theorem 8.5 for oriented graphs.

Theorem 8.9 (Volkmann [154]). Let p�2 be an integer and let D be an oriented graph with clique number �(D)�p.If n(D)�4�(G)p/(p − 1) − 1, then D is maximally local-edge-connected.

Furthermore, Volkmann [154] presented an extension of Corollary 6.5 by Fàbrega and Fiol [45] that each bipartitedigraph of diameter at most 3 is maximally connected.

9. Restricted edge-connectivity

The restricted edge-connectivity �′(G), introduced and studied first by Esfahanian and Hakimi [42] in 1988, is theminimum cardinality over all edge-cuts S in a graph G such that there are no isolated vertices in G−S. The definition of


the restricted edge-connectivity is a special case of a quite general concept of conditional edge-connectivity, proposedby Harary [67] in 1983 (cf. also Section 11). A restricted edge-cut S is called a �′-cut, if |S| = �′(G). Obviously,for any �′-cut S, the graph G − S consists of exactly two components. A connected graph G is called �′-connectedif �′(G) exists. For a graph G, let �(e) = �G(e) = d(u) + d(v) − 2 be the edge degree of the edge e = uv, and�(G) = min{�G(e) : e ∈ E(G)} denotes the minimum edge degree in G.

Of course, �′(G) does not exist for any star and any graph with fewer than four vertices. In fact, Esfahanian andHakimi [42] have observed that these are the only such graphs.

Theorem 9.1 (Esfahanian and Hakimi [42] 1988). Each connected graph G of order n�4, except a star K1,p, is�′-connected and satisfies

�(G)��′(G)��(G). (5)

Recently, Volkmann [159] proved some analogous results for strongly connected digraphs, and he showed thatTheorem 9.1 follows easily from one of his results.

A �′-connected graph G is called �′-optimal or �′-maximal, if �′(G) = �(G). It should be stated that the bound�′(G)��(G) in (5) is tight in the sense that there are infinitely many graphs for which the equality holds. Examplesare complete graphs and the class of n-cubes (cf. [41]). Next we note some simple properties of �′-optimal graphs.

Observation 9.2. If G is a �′-optimal graph, then

�′(G)��(G) + �(G) − 2, (6)

�′(G)�2�(G) − 2, (7)

�′(G)��(G). (8)

Proof. Let e = uv be an edge of G such that d(u) = �(G). This leads to

�′(G) = �(G)��(e) = d(u) + d(v) − 2��(G) + �(G) − 2

and (6) is proved. Because �′(G) = �(G)��(G) + �(G) − 2, inequality (7) is also true. Since G is connected, (8) isvalid for �(G) = 1, and in the case �(G)�2, bound (8) follows from (7). �

Besides the classical edge-connectivity �(G), the restricted edge-connectivity �′(G) recently received much attention(cf. e.g. [41,90,93,173,180]) as a measure of fault tolerance in networks.

We remark that the inequality �′(G) > �(G) implies that G is super-�. There exist some further simple but interestingconnections between �′-optimality, super-edge-connectivity and maximal edge-connectivity.

Observation 9.3 (Hellwig and Volkmann [76] 2005). If G is a �′-optimal graph with �(G)�3, then G is super-�.

Observation 9.4 (Hellwig and Volkmann [76] 2005). If G is a �′-optimal graph, then �(G) = �(G).

Wang and Li [165] have given a sufficient condition for a graph to be �′-optimal, which is, with respect to Observation9.3, an improvement of Corollary 7.3(iii) when ��3.

Theorem 9.5 (Wang and Li [165] 1999). Let G be a �′-connected graph. If dG(u) + dG(v)�n(G) + 1 for all pairsu, v of non-adjacent vertices, then G is �′-optimal.

Hellwig and Volkmann [74,76] have obtained several sufficient conditions for graphs to be �′-optimal. One of themgeneralizes Theorem 9.5 by Wang and Li [165] and is stated as follows.

Theorem 9.6 (Hellwig and Volkmann [74] 2004). Let G be a �′-connected graph. If |N(u) ∩ N(v)|�3 for all pairsof non-adjacent vertices u, v, then G is �′-optimal.


Inspired by the article [74], Balbuena et al. [8] could relax the conditions in Theorem 9.6. The first of their threemain theorems says:

Theorem 9.7 (Balbuena et al. [8] 2006). Let G be a �′-connected graph of minimum degree ��2 and girth g. Ifdm(G)�g − 2, then G is �′-optimal.

The second and third main theorems in [8] present sufficient conditions for graphs with odd girth g and diameterg − 1 to be �′-optimal. Their second main theorem together with Theorem 9.7 leads to an extension of Theorem 9.6.Other generalizations of Theorem 9.6 can be found in the article by Shang and Zhang [136].

Theorem 9.8 (Hellwig and Volkmann [74] 2004). Let G be a �′-connected graph of order n�10. If |N(u)∩N(v)|�2for all pairs of non-adjacent vertices u, v and �(G)�n/2 + 2, then G is �′-optimal.

A family of examples in [74] shows that the condition for �(G) in Theorem 9.8 is best possible, in the sense that�(G)�n/2 + 3 does not imply �′-optimality.

Theorem 9.9 (Hellwig and Volkmann [76] 2005). Let G be a �′-connected graph. If there exists an independent set Iof vertices such that d(v) = �(G) and uv ∈ E(G) for each v ∈ I and u ∈ V (G) − I , then G is �′-optimal.

The class of graphs satisfying the condition in Theorem 9.9 includes the complete multipartite graphs. The next tworesults are generalizations of Fiol’s Corollary 7.13.

Theorem 9.10 (Hellwig and Volkmann [76] 2005). Let G be a �′-connected and triangle-free graph of order n�4with degree sequence d1 �d2 � · · · �dn = �. Then G is �′-optimal when

max{1,�−1}∑i=1

dn−i � max{1, � − 1} · 1

2·(⌊n

2

⌋+ 2 − 4

n − 3

).

Theorem 9.11 (Shang and Zhang [136] 2007). Let G be a connected bipartite graph of order n�4. If

d(u) + d(v)�2

⌊n + 2

4

⌋+ 1

for each pair of vertices u, v ∈ V (G) with d(u, v) = 2, then G is �′-optimal.

Theorem 9.12 (Shang and Zhang [136] 2007). If G is a �′-connected graph that satisfies the following two conditions,then G is �′-optimal.

(a) d(u) + d(v)�2n(G)/2 − 3 for each pair of vertices u, v ∈ V (G) with d(u, v) = 2,(b) each triangle contains at least one vertex w such that d(w)�n(G)/2 + 1.

Theorems 9.11 and 9.12 are improvements of corresponding results by Hellwig and Volkmann [76]. Because ofObservation 9.3, the following result by Wang and Li [166] is an extension of Theorem 7.8 for ��3.

Theorem 9.13 (Wang and Li [166] 2001). If G is a �′-connected graph of minimum degree ��3 and girth g such thatdm(G)�g − 2, then G is �′-optimal.

Theorem 9.14 (Ou [114] 2001). Let G be a �-regular graph with ��2 of order n�4. If � > n/2, then G is �′-optimal.

Ou [114] also exemplified that the lower bound of the minimum degree in Theorem 9.14 is best possible.For X ⊂ V (G) let (X, X) be the set of edges of the graph G with one end in X and the other in X=V (G)−X. In the

case that (X, X) is a �′-cut, the set X is called a �′-fragment. Clearly, if X is a �′-fragment, then X is also a �′-fragment.


Let

r(G) = min{|X| : |X| is a �′-fragment of G}.

A �′-fragment X of a graph G is a �′-atom of G if |X| = r(G).Of course, 2�r(G)�n(G)/2 and G[X] as well as G[X] are connected when X is a �′-atom. The next theorem, due

to Xu and Xu [180], yields a sufficient and necessary condition for a �′-connected graph to be �′-optimal.

Theorem 9.15 (Xu and Xu [180] 2002). A �′-connected graph G is �′-optimal if and only if r(G) = 2.

Theorem 9.16 (Zhang [185] 2007). Let G be a �′-connected graph with �(G)�2. If G is not �′-optimal, thenr(G)��(G)/2� + 1.

Zhang [185] pointed out that the requirement �(G)�2 in Theorem 9.16 is necessary, as can be seen by the followingexample. Let w be a vertex of the complete graph H = Kt+1 with t �4, and let u and v be two further vertices. Nowlet G consist of H, u, v and the two edges uw as well as vw. We observe that �′(G) = t < t + 1 = �(G) and r(G) = 3,and this leads to

r(G) = 3 < 4��(t + 1)/2� + 1 = ��(G)/2� + 1.

Because �(G)�2�(G) − 2, Theorems 9.15 and 9.16 imply immediately the next two corollaries.

Corollary 9.17 (Ueffing and Volkmann [145] 2003). Let G be a �′-connected graph. If G is not �′-optimal, thenr(G)� max{3, �(G)}.

Corollary 9.18 (Xu and Xu [180] 2002). Let G be a �-regular and �′-connected graph. If G is not �′-optimal, thenr(G)��3.

Combining Theorem 9.15 with Turán’s bound (3), one can improve the estimate of r(G) in Corollary 9.17 fortriangle-free graphs.

Theorem 9.19 (Ueffing and Volkmann [145] 2003). Let G be a �′-connected and triangle-free graph. If G is not�′-optimal, then r(G)� max{3, 2�(G) − 1}.

Examples show that Theorems 9.16 and 9.19 as well as Corollary 9.17 are best possible. In [180] Xu and Xu havereceived a property on �′-atoms for non-�′-optimal graphs.

Theorem 9.20 (Xu and Xu [180] 2002). Let G be a �′-connected graph. If G is not �′-optimal, then any two distinct�′-atoms of G are disjoint.

Note that two distinct �′-atoms may not be disjoint in general, as, for example, cycles of length at least four show,which are 2-regular, 2-edge-connected and �′-optimal. However, Meng [108] could prove:

Theorem 9.21 (Meng [108] 2003). Any two �′-atoms of a �-regular and �-edge-connected graph G are disjoint whenr(G)�3.

In order to provide more accurate measures for the fault tolerance of systems of interconnection, Li and Li [93]proposed the concept of super-�′. A graph G is said to be super-edge-connected or shortly super-�′ if every �′-cutof G isolates an edge of G. Obviously, every super-�′ graph is also �′-optimal. However, the cycle Cn with n�6shows that the converse is not true in general. Our first sufficient condition for graphs to be super-�′ is closely relatedto Theorem 9.12.


Theorem 9.22 (Shang and Zhang [136] 2007). Let G be a connected graph of order n�4. If G satisfies the followingtwo conditions, then G is super-�′:

(a) d(u) + d(v)�2n(G)/2 − 1 for each pair of vertices u, v ∈ V (G) with d(u, v) = 2,(b) each triangle contains at least one vertex w such that d(w)�n(G)/2 + 2.

An example by Shang and Zhang [136] shows that the lower bound in condition (b) of Theorem 9.22 is sharp.

Theorem 9.23 (Shang and Zhang [136] 2007). Let G be a connected triangle-free graph of order n�4 and differentfrom the complete bipartite graph K2,n−2. If |N(u) ∩ N(v)|�2 for all pairs u, v of non-adjacent vertices, then G issuper-�′.

Theorem 9.24 (Ou and Zhang [125] 2005). Let G be a �-regular graph. If � > (n(G)/2) + 1, then G is super-�′.

Examples by Ou and Zhang [125] show that the lower bound on � in Theorem 9.24 is best possible. In [120,121],Ou and Zhang proved that undirected de Bruijn graphs UB(d, n) are super-�′ when d �3 and n�4 and that undirectedbinary Kautz graphs UK(2, n) are super-�′ when n�3, respectively. For more information on super-�′ of graphs, pleaserefer to Deng et al. [38], Li and Li [93], Lü et al. [100] and Wang [163].

Further sufficient conditions for graphs to be �′-optimal were given, for example, by Balbuena et al. [5] for graphswith small conditional diameter, Balbuena et al. [10] for permutation graphs, Fan et al. [48] for Kautz undirected graphs,Latifi et al. [90] and Wu and Guo [173] for hypercubes, Li and Li [93] for circulant graphs, Li and Xu [92], Meng [108],Wang [163], Xu [176] and Xu and Xu [180] for edge-transitive and vertex-transitive graphs, Meng [109] for Cayleygraphs on symmetric groups, Ueffing and Volkmann [145] for the Cartesian product of graphs, Xu et al. [178] for deBruijn undirected graphs, Zhang [184] for different graph operations and Zhang and Xu [183] for the n-dimensionalMöbius cube.

10. Maximally connected (di-)graphs

In 1967, Watkins [169] determined the first relationship between order, diameter and connectivity. With the aid ofMenger’s theorem [111], the following result can be established rather easily.

Theorem 10.1 (Watkins [169] 1967). If G is a connected and nontrivial graph, then n(G)��(G)(dm(G) − 1) + 2.

Given positive integers � and dm, Watkins [169] has constructed graphs of order n, with diameter dm and connectivity� such that n = �(dm − 1) + 2.

Later, Kane and Mohanty [86] refined this result by the amount 2(�−�) when dm(G)�3, and they present examples(also in the case � > �), which demonstrate that their bound is best possible.

Theorem 10.2 (Kane and Mohanty [86] 1978). Let G be a connected graph of diameter at least 2. Then n(G)��(G)

(dm(G) − 3) + 2�(G) + 2 when dm(G)�3 and n(G)��(G) + 2 when dm(G) = 2.

Kane and Mohanty proved Theorem 10.2 without using Menger’s theorem. This is somewhat surprising, sinceMenger’s theorem yields Theorem 10.2 as easy as Watkins obtained his Theorem 10.1. In the case that dm(G)�3, thismethod even leads to the slightly better bound n��(G)(dm(G) − 3) + dn + dn−1 + 2, where d1 �d2 � · · · �dn is thedegree sequence of G. Using Menger’s theorem, one can prove Theorem 10.2 also for digraphs (cf. [153, p. 281]). Foranother extension of Watkin’s observation, we refer the reader to Seidman [135].

As would be expected, the higher the degrees of the vertices of a graph, the more likely it is that the graph has largeconnectivity. There are several sufficient conditions of this type. We start with one of the simplest of these, originallypresented by Chartrand and Harary [27].

Theorem 10.3 (Chartrand and Harary [27] 1968). If G is a non-complete graph, then �(G)�2�(G) + 2 − n(G).


This implies that �(G)=�(G) when n(G)��(G)+2. One year later, Bondy [20] generalized Theorem 10.3 as wellas a result of Chartrand et al. [28] by the following degree sequence condition.

Theorem 10.4 (Bondy [20] 1969). Let G be a graph with degree sequence d1 �d2 � · · · �dn. If dn−i+1 � i + k − 1for i = 1, 2, . . . , n − 1 − dk and some integer n − 1�k�1, then �(G)�k.

Another extension of the result in [28] was given by Zamfirescu [181]. In 1971, Geller and Harary [58] have shownthat Theorem 10.3 is also valid for digraphs. This follows from the next more general result.

Theorem 10.5 (Hellwig and Volkmann [79] 2006). Let D be a (di-)graph of order n�4 with degree sequenced1 �d2 � · · · �dn = �. If �(D)�� − k for an integer k with 1�k��, then

�(D)� 1

k + 1

(2k+1∑i=0

dn−i

)+ 2 − n.

Corollary 10.6 (Geller and Harary [58] 1971). If D is a non-complete digraph, then �(D)�2�(D) + 2 − n(D).

Corollary 10.7 (Hellwig and Volkmann [79] 2006). Let D be a (di-)graph of order n�4 with degree sequenced1 �d2 � · · · �dn = �. In addition, let k be an integer with 1�k��. If

2k+1∑i=0

dn−i �(k + 1)(n + � − k − 1) − 1,

then �(D)�� + 1 − k. In particular, if∑3

i=0dn−i �2(n + � − 2) − 1, then D is maximally connected.

Hellwig and Volkmann [79] have given similar conditions also for bipartite (di-)graphs, which lead to a digraphversion of Corollary 10.13 below.

For oriented graphs, Volkmann [156] could improve the bound in Theorem 10.5 as follows.

Theorem 10.8 (Volkmann [156] 2007). Let D be an oriented graph of order n�6 with degree sequence d1 �d2 � · · · �dn = �. If �(D)�� − k for an integer k with 1�k��, then

�(D)� 1

2k + 1

(4k+1∑i=0

dn−i

)+ 2 + 2k − n.

Soneoka et al. [140] established a sufficient condition for maximally connected graphs depending on the diameterand the girth g, and they show that their condition is best possible for girth 4 and odd girth.

Theorem 10.9 (Soneoka et al. [139,140] 1985, 1987). Let G be a graph with girth g. If dm(G)�g − 2 when g is oddor dm(G)�g − 3 when g is even, then �(G) = �(G).

A refinement of Theorem 10.9 can be found in a paper by Fàbrega and Fiol [43]. As a slight extension of a conditionby Esfahanian [40], Soneoka et al. [140] have presented the following result, which is best possible at least for diameters2 and 3.

Theorem 10.10 (Soneoka et al. [140] 1987). Let G be a graph with maximum degree ��3 and minimum degree �.Then �(G) = �(G) when

n(G) > (� − 1)(� − 1)dm(G)−1 + 2.

Theorem 10.11 (Fiol [52] 1993). If G is a �-regular graph with n(G) > �dm(G) + 1, then �(G) = �.


Balbuena and Carmona have given a degree condition for bipartite digraphs to be maximally connected, whichimplies a result by Topp and Volkmann [142].

Theorem 10.12 (Balbuena and Carmona [2] 2001). Let D be a bipartite digraph. If

d+D(u) + d−

D(v)� n(D) + �(D)

2

for all pairs of vertices u, v with dD(u, v)�3, then �(D) = �(D).

Corollary 10.13 (Topp and Volkmann [142] 1993). If G is a bipartite graph of order n(G)�3�(G), then �(G)=�(G).

For �-regular bipartite graphs, Topp and Volkmann have relaxed the condition in Corollary 10.13.

Theorem 10.14 (Topp and Volkmann [142] 1993). If G is a �-regular bipartite graph of order n < 3�+√2� − 1, then

�(G) = �(G).

Theorem 10.15 (Fàbrega and Fiol [45] 1996). If G is a bipartite graph of minimum degree ��2 and girth g such thatdm(G)�g − 2, then �(G) = �(G).

Further bounds for the connectivity as well as for the edge-connectivity in terms of diameter and girth can be foundin the publications by Balbuena et al. [6] and Balbuena and Marcote [9].

As an application of Turán’s inequality (3), we recently obtained a generalization of a theorem by Topp and Volkmann[142] on p-partite graphs.

Theorem 10.16 (Hellwig and Volkmann [78] 2006). Let p�2 be an integer, and let G be a connected graph withclique number �(G)�p. If n(G)��(G)(2p − 1)/(2p − 3), then �(G) = �(G).

Corollary 10.17 (Topp and Volkmann [142] 1993). Let p�2 be an integer, and let G be a p-partite graph. Ifn(G)��(G)(2p − 1)/(2p − 3), then �(G) = �(G).

Examples in [142] show that Corollary 10.17 is best possible for p-partite graphs and thus also for graphs G withclique number �(G)�p. Note that the proof of Theorem 10.16 is completely different from this one of Corollary10.17.

The local connectivity �G(u, v)=�(u, v) between two distinct vertices u and v of a graph G is the maximum numberof internal u–v paths in G. It is a well-known consequence of Menger’s theorem [111] that �(G)=min{�G(u, v)|u, v ∈V (G)}. Obviously, �(u, v)� min{d(u), d(v)}, and we call a graph maximally local connected when �(u, v) =min{d(u), d(v)} for all pairs u and v of vertices in G.

Observation 10.18. If G is maximally local connected, then it is maximally connected.

Proof. Since G is maximally local connected, we have �(u, v) = min{d(u), d(v)} for all pairs u and v of vertices inG. This implies

�(G) = minu,v∈V (G)

{�(u, v)} = minu,v∈V (G)

{min{d(u), d(v)}} = �(G). �

Using Corollary 10.17, Volkmann [158] recently proved the following generalization of this result by Topp andVolkmann.

Theorem 10.19 (Volkmann [158]). Let p�2 be an integer, and let G be a p-partite graph. Then G is maximally localconnected when

n(G)��(G) · 2p − 1

2p − 3.


A graph obtained from a complete graph K4 of order 4 by removing an arbitrary edge is a diamond. We call a graphdiamond-free if it contains no diamond as a subgraph, and we call it C4-free if it contains no cycle C4 of length fouras a subgraph. Now we present a further extension of Corollary 10.13 for ��3 that is best possible.

Theorem 10.20 (Dankelmann et al. [32]). Let G be a connected and diamond-free graph with minimum degree ��3.If n(G)�3�(G), then �(G) = �(G).

Theorem 10.21 (Dankelmann et al. [32]). Let G be a connected and C4-free graph of order n and minimum degree��2. Then �(G) = �(G) when

n�{

2�2 − 3� + 2 if � is even,

2�2 − 3� + 4 if � is odd.

There only exist examples for � = 2 and 3 with equality in Theorem 10.21. However, following a well-knownconstruction (see, for example, [59]) of C4-free graphs, the authors in [32] could show that Theorem 10.21 is at leastasymptotically best possible. It seems to be very difficult to find sharp bounds in general.

Cages were introduced by Tutte [144] in 1947, and since then have been widely studied. Erdös and Sachs [39]proved that (�, g)-cages always exist for any given value of the pair (�, g). Some fundamental properties of cages wereestablished by Fu et al. [57]. They proved that all cages are 2-connected, and they proposed the following conjecture.

Conjecture 10.22 (Fu et al. [57] 1997). Every (�, g)-cage is maximally connected.

The next four results support this conjecture.

Theorem 10.23 (Jiang and Mubayi [84] 1998, Daven and Rodger [37] 1999). All (�, g)-cages are 3-connected for��3.

Theorem 10.24 (Xu et al. [174] 2002). Every (4, g)-cage is maximally connected.

Theorem 10.25 (Marcote et al. [106] 2005). Every (�, g)-cage with either � = 4 or ��5 and g�10 is 4-connected.

Theorem 10.26 (Marcote et al. [105] 2007). If ��2, then every (�, 6)-cage as well as every (�, 8)-cage is maximallyconnected.

In addition, Marcote et al. [105] assure maximal connectivity for some (�, 5)-cages. With the help of Theorem 10.25and a further result by Balbuena et al. [4], Marcote et al. [106] also received the statement of Theorem 10.24.For more information on cages, see the survey by Wong [172], and the Web site maintained by Royle(http://www.cs.uwa.edu.au/∼gordon/data.html).

Mader [101] and Watkins [170] have proved that connected edge-transitive graphs are maximally connected, andBrouwer and Mesner [23] have shown that strongly regular graphs are maximally connected. Imrich [82] and Hamidoune[65] have shown that Cayley graphs and digraphs in certain cases are maximally connected. Li [95] derived an explicitexpression for the connectivity of a connected circulant graph whose connectivity is less than its degree. Liu [98] hasgiven a necessary and sufficient condition for directed circulant graphs to be maximally connected.

11. p-q-Restricted (edge-)connectivity

In 1983, Harary [67] proposed the concept of conditional (edge-)connectivity. The P-edge-connectivity(P-connectivity) �(G, P ) (�(G, P )) of a connected graph G equals the minimum cardinality of a set S of edges(vertices) such that G − S is disconnected and every component of G − S has a given property P.

The study of conditional connectivity has received much attention in the recent years, since it provides a new andinteresting measure for fault tolerance in networks.

http://www.cs.uwa.edu.au/~gordon/data.html


For a positive integer p, Fàbrega and Fiol [46] defined the restricted edge-connectivity �p(G) of a connected graphG as the minimum cardinality of an edge-cut over all edge-cuts S of G such that each component of G − S contains atleast p vertices. Note that G − S has exactly two components for each minimum restricted edge-cut.

In order to find a suitable definition for the restricted (vertex-)connectivity, we notice that G − S can consist ofmore than two components for a minimum vertex-cut S. Since components (except two of them) which do not satisfya given property can be removed by adding them to the vertex-cut, it makes sense to ask for given properties only fortwo components. Following this idea, Hellwig et al. [71] have generalized the conditional connectivity of Harary asfollows.

Definition 11.1. Let P1 and P2 be two graphical properties. The parameter �(G, P1, P2) (�(G, P1, P2)) of a connectedgraph G equals the minimum cardinality of a set S of edges (vertices) such that one component of G − S has propertyP1 and another one has property P2.

In this section, we will consider the special case that one component has at least p vertices and another one at leastq vertices. This leads to the next definition by Hellwig et al. [71].

Definition 11.2. Let G be a connected graph, and let p, q �1 be two integers. An edge-cut (vertex-cut) S of G isa p-q-edge-cut (p-q-vertex-cut), if one component of G − S contains at least p vertices and another one at leastq vertices. The graph is called �p,q -connected (�p,q -connected), if a p-q-edge-cut (p-q-vertex-cut) exists. The p-q-restricted edge-connectivity �p,q(G) (p-q-restricted connectivity �p,q(G)) of a �p,q -connected (�p,q -connected) graphG is the minimum cardinality of a p-q-edge-cut (p-q-vertex-cut) of G. A p-q-edge-cut (p-q-vertex-cut) S of G is calleda minimum p-q-edge-cut (p-q-vertex-cut) or a �p,q -cut (�p,q -cut), if |S| = �p,q(G) (|S| = �p,q(G)).

Note that �1,1(G) = �(G), �1,1(G) = �(G) and �2,2(G) = �′(G). In addition, if S is a �p,q -cut of G, then G − S

consists of exactly two components.

Observation 11.3 (Hellwig et al. [71] 2005). For fixed p and q one can easily determine the values �p,q(G) and�p,q(G) for some graph G in polynomial time by contradicting all choices of disjoint vertex sets of cardinalities p andq that induce connected subgraphs of G and determining minimum sets of edges (or vertices) that separate the twovertices created by the contractions which can clearly be done using max-flow algorithms.

Next we present a sufficient and necessary condition for graphs to be �p,q -connected and �p,q -connected, respectively.

Observation 11.4 (Hellwig et al. [71] 2005). Let G be a connected graph, and let p, q �1 be two integers. The graphG is �p,q -connected (�p,q -connected) if and only if there exist two disjoint vertex sets X, Y with |X|�p and |Y |�q

such that G[X] and G[Y ] are connected (and (X, Y ) = ∅).

For some special cases more transparent characterizations of �p,q -connected as well as �p,q -connected graphs areknown.

Theorem 11.5 (Esfahanian and Hakimi [42] 1988, Bonsma et al. [22] 2002). Let G be a connected graph of order n:

(i) G is �1,q -connected if and only if n�q + 1 and �1,q(G)��(G).(ii) G is �2,2-connected if and only if n�4 and G is not a star (cf. Theorem 9.1).

(iii) G is �3,3-connected if and only if n�6 and G is not isomorphic to the net N or to any graph of the family Fin Fig. 2.

Three years after Bonsma et al. [22], Ou and Zhang [124] as well as Wang and Li [168] have proved statement (iii)in Theorem 11.5 once more. The next observation is easy to prove but useful.

Observation 11.6 (Volkmann [161]). Let p, q �1 be two integers, and let G be a connected graph of order n�p+q+1.Then G is �p,q -connected if and only if G is �p,q+1-connected or �p+1,q -connected.


Fig. 2.

Applying Observation 11.6 and Theorem 11.5(ii) and (iii), we observe that a connected graph G of order n�5 is�2,3-connected if and only if G is not a star and a connected graph G of order n�7 is �3,4-connected if and only if G isnot isomorphic to a member of the family F in Fig. 2. In [161], Volkmann has also characterized all �2,4-, �2,5-, �4,4-and �4,5-connected graphs of order at least 6, 7, 8 and 9, respectively.

Theorem 11.7 (Hellwig et al. [71] 2005). Let G be a connected graph of order n�4. The graph G is not �1,2-connectedif and only if G is a complete multipartite graph.

The �1,3-connected graphs are also characterized in [71]. The next result is an extension of Whitney’s classicalinequality �(G)��(G).

Theorem 11.8 (Hellwig et al. [71] 2005). If G is a �1,q -connected and �q,q -connected graph, then

�1,q(G)��q,q(G).

Examples in [71] show that �1,q(G)��q−1,q(G) is not true in general. The following example demonstrates that�2,q(G)��r,s(G) with 2�q �r, s and r + s� |V (G)| is not always true.

Example 11.9 (Hellwig et al. [71] 2005). Let r, s be integers with r > 8 and s�2. Let B be a complete bipartite graphK3,r−5 with partite sets {xi : i = 2, 3, . . . , r − 4} and {a, b, c}, and let B ′ be a complete bipartite graph K2,s−2 withpartite sets {yi : i = 1, 2, . . . , s − 2} and {u, v}.

We define the vertex set of the graph H as the disjoint union of the vertex sets of B and B ′ together with two additionalvertices x1 and xr−3. The edge set of H contains the edge sets of B and B ′ and the edges x1a, x1b, xr−3b, xr−3c, ba, bc

and uv. Furthermore, we join the vertices a, b and c with u and v by new edges.It is easy to see that �r,s(G)� |({a, b, c}, {u, v})|= 6. The unique possibility for a 2-2-vertex-cut is to disconnect the

edges x1a and xr−3c. This requires the removal of the vertices u, v, b and xi for i = 2, . . . , r − 4, which implies that�2,2(H) = r − 2 > 6 and hence �2,2(H) > �r,s(H).

Inspired by Whitney’s inequality (1), Harary [67] asked in 1983 if for any graphical property P of a graph G theinequality

�(G, P )��(G, P )

is true. In 2005, Hellwig et al. [71] have used Example 11.9 as follows, to give a negative answer to Harary’s question.If P denotes the property that a graph contains at least two vertices, then we conclude from the definitions that

�2,2(G)��(G, P ) and �2,2(G) = �(G, P ) for each graph G. Hence, it follows for the graph H in Example 11.9 that

�(H, P )��2,2(H) > �r,s(H)��2,2(H) = �(H, P ).

If G is a �1,2-connected graph, then the inequality �1,2(G)�n(G)−3 is immediate. Using an explicit characterizationof claw- and paw-free graphs, given by Faudree et al. [49], we could determine all graphs with �1,2(G) = n(G) − 3.


Theorem 11.10 (Hellwig et al. [71] 2005). Let G be a �1,2-connected graph. The graph G satisfies �1,2(G)=n(G)−3if and only if the complementary graph G is claw-, paw-, diamond- and C4-free.

The following strong result by Gyori [64] and Lovász [99] easily leads to the first two interesting sufficient conditionsin Theorem 11.12 for graphs to be �p,q -connected.

Theorem 11.11 (Gyori [64] 1978, Lovász [99] 1977). For every k-connected graph G of order n, k vertices v1, v2, . . . ,

vk ∈ V (G) and k positive integers n1, n2, . . . , nk such that n1+n2+· · ·+nk=n, there exists a partition {V1, V2, . . . , Vk}of V (G) such that vi ∈ Vi , |Vi | = ni and G[Vi] is connected for 1� i�k.

Theorem 11.12 (Rautenbach and Volkmann [130]). Let p and q be integers with q �p�1. A connected graph G oforder n�p + q and minimum degree � is �p,q -connected provided one of the following conditions is satisfied:

(i) G is 2-connected.(ii) G has a block of order at least p + 1 containing at most one cut vertex.

(iii) p = q �� + 1 and G contains a block with at least two cut vertices.(iv) n�2q − 1 and G contains a cut vertex u such that all components of G − u are of order at least p.

In [130], the authors study �2,q -connected graphs in detail. For example, they characterized the �2,q -connected treesas follows.

Theorem 11.13 (Rautenbach andVolkmann [130]). Let q �2 be an integer. A tree T of order n�q+2 is �2,q -connectedif and only if it contains a non-endvertex u of degree at most n − q which is adjacent to at most one non-endvertex.

Using results about cyclic sums, Rautenbach and Volkmann [131] derived sufficient conditions for a graph of largeenough order containing a cycle long enough to be �p,q -connected.

Theorem 11.14 (Rautenbach and Volkmann [131]). Let p, q, r be positive integers with r �3 and p + q �2r − 1. IfG is a connected graph of order n�p + q which contains a cycle of length r, then G is �p,q -connected.

The next theorem generalizes the main result of Ou [117]. Note that the proof works along the same lines as theproof in [117] but that the authors in [132] present a considerably shorter argument.

Theorem 11.15 (Rautenbach and Volkmann [132]). Let p, q be integers with 2�p�q, and let G be a connectedgraph of order n� max{2q − 1, 3p − 2}. Then G is �p,q -connected if and only if G contains no cut vertex u with theproperty that each component of G − u has at most p − 1 vertices.

Choosing p = q in Theorem 11.15, we obtain immediately the above-mentioned main result of Ou [117].

Corollary 11.16 (Ou [117] 2005). Let G be a connected graph of order n�3p − 2. Then G is �p,p-connected if andonly if G contains no cut vertex u with the property that each component of G − u has at most p − 1 vertices.

In [132], the reader can find further applications of Theorem 11.11. In [131,132], Rautenbach and Volkmann alsostudy �p1,p2,...,pk

-connected graphs. For positive integers p1, p2, . . . , pk we say that a connected graph G is �p1,p2,...,pk-

connected if it has an edge-cut S ⊆ E(G) such that G − S has k components with vertex sets V1, V2, . . . , Vk such that|Vi |�pi for 1� i�k.

However, now we will discuss the case p = q more in detail, and thus we write in the following again �p(G) insteadof �p,p(G).

Recently, Ou [117] has given the following sufficient and necessary conditions for graphs to be �p-connected.

Theorem 11.17 (Ou [117] 2005). Let G be a connected graph of order n�2p. Then G is �p-connected if and only ifG has a spanning tree T such that T − u has a component of order at least p for any vertex u ∈ V (T ).


Fig. 3.

Following [22,110,115], we define an extension of the minimum edge degree �(G) of a graph G for an integerp�2 by

�p(G) = min{|(X, X)| : X ⊂ V (G), |X| = p, G[X] is connected}.Notice that �2(G)=�(G). For p=3, Bonsma et al. [22] gave an inequality analogous to �2(G)��2(G) of Esfahanian

and Hakimi [42] in Theorem 9.1.

Theorem 11.18 (Bonsma et al. [22] 2002). If G is a �3-connected graph, then �3(G)��3(G).

Three years later, Wang and Li [168] have also proved the inequality �3(G)��3(G) for �3-connected graphs. Inaddition, Meng and Ji [110] said ‘It is easy to see that �3(G)��3(G)’. However, we think that this inequality is notimmediate.

Remark 11.19 (Bonsma et al. [22] 2002). For p�4, the inequality�p(G)��p(G) is no longer true in general. Let G bethe disjoint union of a complete graph Kp and the vertices y1, x1, x2, . . . , xp−1 together with the edges yy1, xx1, xixi+1

for 1� i�p − 2, where x, y ∈ V (Kp) (cf. Fig. 3). Then �p(G)=|(V (Kp), V (Kp))|= 2 and �p(G)=p − 1 > �p(G).

However, in special cases this inequality is also true for p�4, as the next two results will demonstrate.

Theorem 11.20 (Ou and Zhang [119] 2003). Let G be a �-regular and �p-connected graph. If the girth g�(p/2)+2,then �p(G)��p(G).

Let H1, H2, . . . , Hr be r copies of the complete graph Kt and let u be a further vertex. If we connect the vertex u

with every vertex of V (Hi) for i = 1, 2, . . . , r by an edge, then we denote the resulting graph by Hr,t .

Theorem 11.21 (Zhang andYuan [187] 2005). Let G be a connected graph of minimum degree � and order n�2(�+1)

which is not isomorphic to any Hr,�. Then G is �p-connected for any p�� + 1 and �p(G)��p(G).

If (X, X) is a �p-cut, then X ⊂ V (G) is called a �p-fragment. Let

rp(G) = min{|X| : |X| is a �p-fragment of G}.A �p-fragment X of a graph G is a �p-atom of G if |X| = rp(G). Obviously, p�rp(G)� |V (G)|/2.

If G is a �3-connected graph, then �3(G)��3(G), by Theorem 11.18. A �3-connected graph G is called �3-optimalif �3(G) = �3(G). The following characterization of �3-optimal graphs was inspired by Theorem 9.15 of Xu and Xu[180].

Theorem 11.22 (Bonsma et al. [22] 2002). A �3-connected graph G is �3-optimal if and only if r3(G) = 3.

The next result by Zhang [185] is an analogue to Theorem 9.16 for �3-connected graphs.

Theorem 11.23 (Zhang [185] 2007). Let G be a �3-connected graph with �(G)�3. If G is not �3-optimal, thenr3(G)��3(G)/3� + 1.


Examples by Zhang [185] demonstrate that the requirement �(G)�3 in Theorem 11.23 is necessary. Because�3(G)�3�(G) − 6, Theorems 11.22 and 11.23 imply immediately the next corollary.

Corollary 11.24 (Bonsma et al. [22] 2002). Let G be a �3-connected graph. If G is not �3-optimal, then r3(G)� max{4, �(G) − 1}.

In view of Theorems 9.1, 9.15, 11.18 and 11.22, the next result by Zhang and Yuan [188] is an extension of Corollaries9.17 and 11.24.

Theorem 11.25 (Zhang and Yuan [188] 2007). Let G be a �p-connected graph. If �p(G) < �p(G), then rp(G)�max{p + 1, �(G) + 2 − p}.

Using Theorems 11.5(iii) and 11.18, Wang [164] recently presented an Ore-type condition for graphs to be �3-optimal,which is an analogue to Theorem 9.5 by Wang and Li [165] for �2-connected graphs.

Theorem 11.26 (Wang [164] 2006). If G is a graph of order n�6 such that d(u) + d(v)�n + 3 for any pair ofnon-adjacent vertices u and v, then G is �3-optimal.

Examples by Wang [164] show that Theorem 11.26 is best possible in the sense that the condition d(u)+d(v)�n+2for any pair of non-adjacent vertices u and v does not guarantee that G is �3-optimal. With similar methods, Ou [118]could relax the condition in Theorem 11.26 for triangle-free graphs.

Theorem 11.27 (Ou [118]). If G is a triangle-free graph of order n�6 such that d(u) + d(v)�n − 1 for any pair ofnon-adjacent vertices u and v, then G is �3-optimal.

Conjecture 11.28 (Ou [118]). If G is a connected triangle-free graph of order large enough such that d(u) + d(v)�(n/2) + 2 for any pair of non-adjacent vertices u and v, then G is �3-optimal.

More information on �p-connected graphs can be found in the articles by Meng and Ji [110], Ou [116], Ou andZhang [122,123], Wang and Li [167], Zhang and Meng [186] and Zhang and Yuan [188].

12. Line graphs

The line graph L(G) of a graph G has the edge set E(G) as vertex set and two vertices in V (L(G)) are adjacentwhenever they are incident as edges in G. It is easy to see that |V (L(G))|=|E(G)|, �(L(G))=�(G) and �(L(G))=�(G)

when n�3. Further properties of line graphs can be found in the survey article of Prisner [129].In 1969, Chartrand and Stewart [30] presented the following bounds on the connectivity of line graphs.

Theorem 12.1 (Chartrand and Stewart [30] 1969). If G is a connected graph, then

(i) �(L(G))��(G) when �(G)�2,(ii) �(L(G))�2�(G) − 2,

(iii) �(L(L(G)))�2�(G) − 2.

Especially, Theorem 12.1(i) and Whitney’s bound (1) lead to �(L(G))��(G) when �(G)�2. In connection withthis inequality, Bauer and Tindell [12] have shown that, if a and b are integers such that 1 < a < b, then there exists agraph G with �(G) = a and �(L(G)) = b. This answers a question of Capobianco and Molluzzo [24, p. 65]. Bauer andTindell [12] also established a similar result for the edge-connectivity.

With the help of the next general theorem we could improve the inequalities (i) and (iii) of Theorem 12.1, and theseimprovements lead to a short proof of the bound (5) by Esfahanian and Hakimi [42].

Theorem 12.2 (Hellwig et al. [70] 2004). Let q, p be integers satisfying q �p�2, and let G be a �p,q -connectedgraph. Then L(G) is �p−1,q−1-connected with

�p−1,q−1(L(G))��p,q(G).


If L(G) is �(p2 ),(

q2 )-connected, then

�p,q(G)��( p2 ),( q

2 )(L(G)).

The special case p = q = 2 in Theorem 12.2 implies the following better bound of �(L(G)) in Theorem 12.1(i).

Corollary 12.3 (Hellwig et al. [70] 2004). If G is an arbitrary �2,2-connected graph, then

�1,1(L(G)) = �(L(G)) = �2,2(G)��(G).

Now we can prove the inequality �2,2(G)��(G) of Theorem 9.1 by Esfahanian and Hakimi [42] as follows.If G is an arbitrary �2,2-connected graph, then Corollary 12.3 implies

�2,2(G) = �(L(G))��(L(G)) = �(G).

For a refinement of Corollary 12.3 we refer the reader to [179]. Combining Corollary 12.3 with (i) and (ii) of Theorem12.1, we immediately obtain an extension of Theorem 12.1(iii).

Corollary 12.4 (Hellwig et al. [70] 2004). If G is an arbitrary �2,2-connected graph, then

�(L(L(G)))�2�(G) − 2.

The identity �(G) = �(L(G)) also yields the following characterization of maximally connected line graphs.

Corollary 12.5 (Hellwig et al. [70] 2004). If G is a �2,2-connected graph, then �(L(G)) = �(L(G)) if and only if�2,2(G) = �(G).

A �′-connected graph is called super-�′ if every minimum �′-cut isolates an edge.

Theorem 12.6 (Xu et al. [179] 2005). Let G be a �2,2-connected graph with minimum degree ��3. If G is super-�′,then L(G) is super-�.

Theorem 12.7 (Balbuena et al. [3] 1996). Let G be a connected graph:

(i) If dm(L(G))�3, then G is maximally edge-connected.(ii) If dm(L(G))�2, then G is maximally connected.

Because dm(L(G))�dm(G) + 1 (see, for example, [88]), Theorem 12.7(i) is an generalization of Plesník’sTheorem 3.3.

Theorem 12.8 (Balbuena et al. [3] 1996, Carmona and Fàbrega [25] 1999). Let G be a connected graph of minimumdegree ��2 and girth g:

(i) If dm(L(G))�g when g is odd and dm(L(G))�g − 1 when g is even, then �(G) = �(G).(ii) If dm(L(G))�g − 1 when g is odd and dm(L(G))�g − 2 when g is even, then G is super-� and �(G) = �(G).

Theorem 12.9 (Meng [107] 2001). If G is a 3-connected and regular graph, then L(G) is super-edge-connected.

Further bounds for the connectivity and edge-connectivity for the line graph L(G) and the total graph T (G) weredetermined by Bauer and Tindell [13] and Zamfirescu [182]. Especially, Bauer and Tindel proved that the total graphT (G) of any connected graph is maximally edge-connected. Sun [141] could extend some results by Chartrand andStewart [30] and Zamfirescu [182]. In particular, he has presented the following results.


Theorem 12.10 (Sun [141] 1986). Let G be a connected nontrivial graph with independence number (G):

(i) If �(G) < �(G), then �(L(G)) = �(G).(ii) If (G) < �(G), then L(G) is maximally edge-connected.

In the case that �(G) < �(G), it follows that �2,2(G)=�(G) and thus Theorem 12.10(i) is an immediate consequenceof Corollary 12.3.

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