the threshold weight of a graph

The Threshold Weight of a Graph -

Chi Wang A. C. Williams

RUTCOR-RUTGERS CENTER FOR OR RUTGERS UNIVERSITY

NEW BRUNS WICK, NEW JERSEY

ABSTRACT

The threshold weight of a graph G is introduced as a measure of the amount by which G differs from being a threshold graph. The threshold graphs are precisely the graphs whose threshold weights are 0. At the opposite extreme is the class of graphs for which the threshold weight is the maximum possible. Such graphs are defined as heavy graphs. Among the results are as following: A theorem that specifies the threshold weight of any trianglefree graph; necessary and sufficient conditions for a heavy graph in terms of the solvability of a system of linear inequalities; some sufficient conditions for a graph to be heavy and a necessary condition (conjectured to be sufficient, as well) for a heavy graph in terms of its cliques.

1. INTRODUCTION

Let G = (V, E) be a graph of vertex set V (or V(G)) and edge set E (or E(G)). A graph is called a threshold graph if nonnegative weights can be as- signed to the vertices such that a subset S of V is stable if and only if the sum of weights on S is no larger than a certain threshold value. Threshold graphs were introduced and characterized by Chvital and Hammer [ S ] . Since then, many other characterizations and various generalizations have been studied. A number of such papers are included in the References. In the context of measuring how “nonthreshold” a given graph is, various measurements have been studied. Chvkal and Hammer [5 ] introduced the threshold dimension; Hammer, Ibaraki, and Simeone [18] introduced the threshold gap; while Peled and Simeone [37] introduced the threshold measure. Other discussions of nonthresholdness can be found in [ 5 , 8 , 9 , 11, 15, 21, 22, 27, 30, 42, and other references cited therein]. In this paper, still in the interests of measuring the “nonthresholdness” of a graph and motivated

Journal of Graph Theory, Vol. 15, No. 3, 235-249 (1991) 0 1991 John Wiley & Sons, Inc. CCC 0364-9024/91/030235-15$04.00

236 JOURNAL OF GRAPH THEORY

by the concept of the threshold order introduced by us in [41], we formulate another measure of nonthresholdness, by means of the weight assignment on both vertices and edges. We call this measure the threshold weight of a graph.

First recall some terminologies and definitions. All graphs considered here are finite and undirected. A subset S of the vertex set V of G is stable if no two vertices in S are adjacent in G. A subgraph is an induced subgraph, induced by a subset V ' of V , if it has V ' as its vertex set and edge set of all edges of G with both ends in V'. An edge-induced subgraph induced by a subset E' of the edge set E of G is an induced subgraph of G induced by the vertex set V ' consisting of all of the ends of edges in E'. A graph is k-partite if the vertex set V of G can be partitioned into k disjoint subsets K , V2, . . . , vk such that no vertices in the same K are adjacent. A complete k-partite graph has an edge between every pair of vertices that are not from the same K . (A 2-partite graph is also called a bipartite graph.) A graph is called triangle-free if it contains no triangle as an induced subgraph. Other terminologies and definitions not defined here are taken from [4].

In contrast to other approaches to thresholdness, which assign weights only to the vertices, we assign weights to both vertices and edges. Formally, given any graph G = (V, E), we assign a positive threshold value t to the graph, nonnegative weights wi to each vertex i , and nonnegative weights we to each edge e, such that for any subset S of V , S is a stable set of G if and only if the total weight of the vertices and of the edges over the subgraph G(S) induced by S is less than the threshold value, i.e.,

1 wi + c w, < t , iff S is a stable set. (1) iES eEG(S)

The threshold weight W(G) is defined as the optimal value of the following linear programming problem:

s.t. c w i I t - 1, for each maximal stable set S G V , iES

Wi + W, + We 2 t, for each e = ( i , j ) E E ,

w 10. (2)

An assignment of weights to the vertices and edges of G is called a feasible threshold assignment if it is feasible for (2). It is an optimal threshold assignment if it is an optimal solution in (2). It is clear that if (w; t ) is a feasible threshold assignment for G, then for any induced subgraph H of G, (w, t ) restricted to H is also a feasible threshold assignment for H.

The linear programming problem (2) is always feasible with optimal value bounded below by 0 and above by (E(G)I, the number of edges of G,

THRESHOLD WEIGHT 237

since t = 1, wi = 0 V i , we = 1 V e is feasible. The optimal value of (2), W(G), equals 0 if and only if G is a threshold graph. Therefore, when the optimal solution is greater than zero, i.e., when the graph G is not a threshold graph, W(G) may be interpreted as a measure of the nonthresholdness of G.

The concept of the threshold weight introduced here is motivated by the work on the so-called threshold order of a Boolean function, developed by the current authors in [41]. We showed that separating all of the stable sets of a graph G from the nonstable sets can always be done by a quadratic sur- face, so all graphs are of threshold order at most 2. By requiring the separating polynomial to have non-negative coefficients, i.e., to be a positive separator as defined in [41], the threshold weight defined here is equal to the minimum of the total weight on the quadratic terms.

2. A N INTERPRETATION

Consider a situation in which a number of computer users, or processors within a parallel processing computer, are tied together in a network capa- ble of broadcasting and receiving data and other messages. Each user i wishes to communicate with a subset of some of the other users, but not with others. The problem is to devise a system whereby each user’s software can distinguish quicky between messages to be accepted and those to be rejected as they arrive on the broadcast network.

One possible solution is to store a list of acceptable message origins, i.e., to store at each user i’s site the list of users j that user i is communicating with. This assumes memory, table look-up capability, and perhaps consid- erable compute time to field all the messages arriving. Better would be a protocol whereby each message from user i contains as a header a small integer code number wi such that userj accepts a message from user i if and only if wi is greater than or equal to wj plus some threshold value t. Minimal computational capability is needed and the processing would be fast.

Consider the graph G(V, E) where the vertices I/ are the users and where an edge eij is included in G whenever user i can communicates with userj. The simple protocol can be implemented only if G is a threshold graph. Otherwise we can supplement the code numbers wi, by numbers wi,, depending on both i and j, so that user j accepts a message if and only if wi + wij is greater than or equal to a t - wj, where t is certain threshold value. One possible implementation is to have the header for each message from user i to consist of wi, followed by a list of indices j k for which wijk is not zero, followed by the value wvk. If the list j k is short, the determination of whether or not to accept the message would still be relatively simple and fast.

If the code numbers are to be chosen so as to minimize the total bits re- quired to represent the wij, the problem is exactly the linear program (2) for the threshold weight. Other criteria, e.g., minimize total header lists set on


the network, perhaps taking account of the relative frequencies of messages from each i to eachj, could generally be represented by variants of (2 ) .

3. SOME PRELIMINARY LEMMAS

As mentioned, threshold graphs are graphs of threshold weight 0. A number of characterizations have been given. One of the characterizations by means of forbidden induced subgraphs [5] is as follows: Let i , j , k , 1 be dis- tinct vertices in a graph G, such that ( i , j ) , ( k , I ) are two edges of G and there is no edge between vertex i and vertex k , and there is no edge between j and 1. We say that ( i , j ) , ( k , I ) are independent edges. (Figure 1).

Theorem 3.1. only if G has no independent edges.

(Chvital and Hammer [5]) . A graph G is threshold if and

Let P4, C4, and 2 K 2 be the graphs shown below in Figure 2 . Theorem 3.1 says that a graph is a threshold graph if and only if it does not contain P4, C4, or 2 K 2 as an induced subgraph.

Therefore, the threshold weights on an independent pair of edges cannot be zero in a nonthreshold graph.

Lemma 3.2. Let e = (i, j ) , f = (k, I) E E be two independent edges in a graph G = (V, E). Then W , + wr 2 2 in any feasible threshold assignment for G.

ProoJ: Let (w, r ) be a feasible threshold assignment for graph G; then

The lemma follows from

Note that Lemma 3.2 holds without requiring that w 2 0.

independent edges

FIGURE 1


Corollary 3.3. If a graph G is threshold, then W(G) = 0. If G is not threshold, then W(G) 1 2. There is no graph G such that 0 < W(G) < 2.

Proof: If G is threshold graph, then W(G) = 0. If not, G has at least one pair of independent edges e and f. By Lemma 3.2, W(G) 2 w, + ~f 2 2. QED.

Lemma 3.4. Let (w; t) be a feasible threshold assignment for the graph G. Then the edge-induced subgraph G(L), induced by edge set L = {e E V(G) I we < l}, is an induced threshold subgraph of G.

ProoJ If (w; t) is a feasible threshold assignment for G, it is a feasible assignment for any induced subgraph of G when restricted to the subgraph. Also, by the definition of L and Lemma 3.2, G(L) cannot contain any independent edges, otherwise there would be a pair of independent edges with at least one of them having weight at least 1. By Theorem 3.1, therefore, G(L) is a threshold graph. QED.

4. HEAVY GRAPHS

From here on, we discuss graphs that are "most nonthreshold" in the sense that their threshold weights reach their upper bounds in formulation (2). Such graphs are at the opposite extreme from the threshold graphs. We call a graph heavy if W(G) = IE(G)I, i.e., if its threshold weight is the maximum possible. It is clear that there can be no characterization of heavy graphs in terms of forbidden induced subgraphs, since the heavy property is not necessarily inherited. In fact, any nonempty graph has some nonempty induced threshold subgraphs. But we have the following necessary and sufficient condition for heavy graphs.

Theorem 4.1. A graph G is a heavy graph if and only if the following linear inequality system is unsolvable:


c z i I 1, for each maximal stable set S of V, IES

z 1 0 . (4)

where d , is the degree of vertex i, i.e., the number of edges incident with i.

ProoJ: The dual of (2) is

C ys - C ye 2 0 , Vi E V, SllES e 1 IEe

y p I 1, V e E E ,

y 2 0 .

Therefore, G is heavy if and only if the following linear inequality system is solvable:

C Y S = IEl S

ys 2 d , , V i E V , SllES

y 2 0 .

By Farkas’s Lemma [12], system (8) is solvable if and only if the following linear inequality system is unsolvable:

dizi + lEla = 1 , i E V

x z i + a I 0,

z 2 0 .

VS maximal stable, iES

Equivalently, the following system, equivalent to the system of (3) and (4), is unsolvable.


c z i s 1, VS maximal stable, iES

ZO,Z L 0. QED.

The following corollaries of Theorem 4.1 give some sufficient conditions for heavy graphs. A graph G is called regular if every vertex of G has the same degree. Let x(G) be the chromatic number, i.e., the minimum number of stable sets needed to cover V(G) . We have

Corollary 4.2. Let G be a regular graph. If x(G) 5 i lVl, then G is a heavy graph.

Mf. Suppose that G is r-regular, i.e., every vertex of G has degree r. Let z be such that (4) holds, i.e., &zi 5 1 for any maximal stable set S of G. Let S1, Sz, . . . Sx(c) be a partition of Vinto x(G) stable sets. Let n = IVI. Then

1=1 i= l

If x(G) I fn , then rX(G) I f r n = IE(G)I. So the system of (3) and (4) is unsolvable. By Theorem 4.1, G is a heavy graph. QED.

Observation 1. The (k - 1)m-regular complete k-partite graph Km,m,... ,m is heavy.

Let 0 be the clique number of G, i.e., the minimum number of cliques needed to cover V(G). A graph is called weaklyperfect if x(G) = 0(G).

Corollary 4.3. then G is a heavy graph.

If a regular graph G is weakly perfect and 8(G) 5 f IV(,

Corollary 4.4. If the vertex set of the graph G can be partitioned into stable sets S1, S1, . . . s k , such that vertices in the same stable set Si have the same degree ki and each stable set has size ISi] at least two, then G is a heavy graph.

Prouf. Let z be such that (4) holds, i.e., XiESzi 5 1 for any maximal stable set S of G. Then

dl + d2 + ... + d, 2 = IEl

So the system of (3) and (4) is unsolvable. By Theorem 4.1, G is a heavy graph. QED.


Observation 2. The complete k-partite graph G,,,,,,...,,k with k 1 1 is heavy if mi 1 2 for every i.

The following is an operation that enables us to construct a heavy graph from a smaller heavy graph. For any vertex x E V of a graph G = (V, E), the neighborhood of x is N ( x ) = { y I (x , y ) E E}. Let y be a new vertex and let Ey be a subset of { ( y , z ) I z E N(x)}. The graph Gx.y = (K,,,E,,,) is defined as K,, = V U {y}, Ex,, = E U Ey. GX,, is said to be an augment of G.

Theorem 4.5. If G is a heavy graph, then any augment G,,, is a heavy graph.

Proof. Suppose z 1 0, CiEszi I 1 for each maximal stable set S of GI,,. Let 2: = z, + z,, zl = zi for i f x,y . Then XiEszf I 1 for each maximal stable set S’ of G. Let di be the degree of vertex i in GX,,. Then the degree d! of i in G is d! = di if i e N(x) ; and d! = di - 1 if i E N(x). Let V = K,,, N(x) = V - {x,y} - N(x). Since G is heavy, we have

Therefore, by Theorem 4.1, G,, is a heavy graph. QED.

An edge of any graph is called appendant if one of its two ends is of degree 1.

Corollary 4.6. Adding appendant edges to any nonisolated vertex of a heavy graph produces a heavy graph.

Corollary 4.7. Every forest with at least two nontrivial connected compo- nents, i.e., neither of which is an isolated vertex, is a heavy graph.

Proof. By Corollary 4.2, mKz is heavy for m 2 2. Since any tree can be obtained by repeated augments starting from a single edge K 2 , we can grow the forest from some mK2. Then the corollary follows from Corollary 4.6. QED.

5. THE THRESHOLD WEIGHT OF TRIANGLE-FREE GRAPHS

In this section, we show that triangle-free graphs, with a few exceptions, are heavy graphs, by giving the threshold weight of all triangle-free graphs. In particular, the threshold weights of bipartite graphs are determined.


A connected graph is called a star if it has at most one vertex of degree larger than 1. A connected graph is called a bi-star if it has exactly two vertices of degree larger than 1 (see Figure 3).

Lemma 5.1. If G is a star, then W(G) = 0. If G is a bi-star, then W(G) = IEl - 1.

proof. If G is a star, it is threshold. So W(G) = 0. Now suppose that G is a bi-star. Let a,b be the two vertices of degree larger than one, where u has m neighbors and 6 has n neighbors (m, n 2 1). An optimal solution of the dual pair (2) and (5), (6), (7) is given as follows. For (2), take we = 1 for each edge e, except that e,b = 0; wi = 0 for each vertex i, except that w, = wb = 1; t = 2. Observe that there are only 3 maximal stable sets: P = N(a), Q = N(6) , R = V - {u,b}. Clearly this is a feasible solution with value m + n. For the dual problem, take ye = 1 for each edge e, except thaty,b = O ; y f = ~ ; Y Q = m;yR = 0. Clearly this is a dual feasible solution with value m + n. QED.

Theorem 5.2. Let G be a triangle-free graph. If G is a star, then W(G) = 0; if G is a bi-star, then W(G) = IEl - 1; if G is neither a star nor a bi-star, then W(G) = (El, i.e., G is a heavy graph.

Proof. Suppose (w; t) is an optimal solution of (2) for G. By Lemma 3.5, the edge-induced subgraph G(L) induced by L = {e 1 we < 1) is an induced threshold subgraph of G. It is clear that the only threshold triangle-free graph without isolated vertices is a star, so that G(L) must be a star. If G(L) has no edges at all, then G is clearly heavy. If G(L) has only one edge e, since G is triangle-free and is not a star or a bi-star, there is at least one edge$ which forms an independent edge pair withe in G. By Lemma 3.2, the total weights on we + wf is at least 2. The weight on every other edge is at least 1 since they are not in G(L). So W(G) 2 [El, i.e., G is a heavy graph.

Now, suppose that the star G(L) has more than one edge, so that G(L) has a unique vertex a of degree larger than 1. Since G is triangle-free, one of following holds:

Case I. a is in some cycle C of length 4 of G.

SIaT bi-slor

FIGURE 3


Case 2. a is in some cycle C of length 5 of G, but not in any cycle of length 4 of G.

Case 3. a is not in any cycle of length shorter than 6 in G, but there are two paths 11, l 2 such that each of them has at least 3 vertices, starts from a, and 11,12 intersect only at a.

Case 4. a is not in any cycle of length shorter than 6 in G and there are no paths as described in Case 3.

Case 1. Let b be the vertex in C not adjacent to a. Then b is not in V(G(L)). Let i = 1,. . . , rn be the vertices of G(L) that are not adjacent to b. Then G has an induced subgraph H as shown in F1 (Figure 4) containing all edges of G(L). By Observation 2, a complete bipartite graph G,,,,, is heavy if m l I 2 and m2 L 2. By Corollary 4.6, adding appendant edges to a nonisolated vertex keeps a heavy graph heavy, so H is heavy. W ( H ) = IE(H)I and W(G) L W ( H ) + (E(G) - E ( H ) ( L (E(G)I, so G is heavy.

Fl

b I

b I

c Y

F3

FIGURE 4


Case 2. There are two vertices x,y in C that are not adjacent to a. Let b, c be the two vertices in C such that b is adjacent a and x , and c is adjacent to a and y. Then x , y cannot be adjacent to any vertex of G(L) other than b or c (b and c may or may not be in G(L)). Otherwise a is in some cycle of length 4. Let i = 1,. . . , m be the vertices of G(L) that are not adjacent to x , y ; then G has an induced subgraph H as shown in Fz (Figure 4) containing all edges of G(L). Suppose z > 0 ZiESzi 4 1 for each maximal stable sets S of V ( H ) . Then we have

Therefore,

Therefore, by Theorem 4.1, H is heavy, W ( H ) = IE(H)I and W(G) 2 W ( H ) + IE(G) - E ( H ) ( 1 IE(G)I. So G is heavy.

Case 3. Let the path l I have ends x and a, and let 12 have ends y and a. Let b be the vertex in l l adjacent to a and x, and let c be the vertex in l2 adjacent to a and y. Then x , y cannot be adjacent to any vertex of G(L) other than b or c (b and c may or may not be in G(L)). Otherwise a is in some cycle of length less than 5. Let i = 1,. . . , m be the vertices of G(L) that are not adjacent tox,y. Then, G has an induced subgraph H as shown in F3 (Figure 4) containing all edges of G(L).

Case 4. Since G is not a star or bi-star, if G has more than one nonsingleton connected component, then G has an induced subgraph K 2 U G(L). By Corollary 4.7, Kz U G(L) is heavy. Therefore so is G.

If G has only one nonsingleton connected component, then there must be an induced path 1 of 4 vertices, a, b, c, d, starting from a. Then b, c, d are not adjacent to any other vertices of G(L) than a, b (b may or may not be in G(L)) . So G has an induced subgraph H as shown in F4 (Figure 4) containing all edges of G(L).

Proceeding as in the proof of Case 2, it can be shown that a graph consisting only of a path of five vertices is heavy. Then by Corollary 4.6,


adding appendant edges to any vertex of that path produces a heavy graph. So both H and G in Cases 3 and 4 are heavy. Hence, if G is neither a star nor a bi-star, G is heavy. QED.

6. CLIQUES AND SPLIT GRAPHS

In this section, we give a necessary condition for heavy graphs and we conjecture that it is also sufficient. Given a subgraph H of G, denote G\H be the graph obtained from G by removing all vertices of H and all edges with an end in V ( H ) .

Theorem 6.1. If G is a heavy graph, then for each clique of G, IE(G\C)I 2

IE(C)l.

Proof. Let C be a clique such that IE(G\C)I < IE(C)). Let the edges of C be el,e2, . . . ,er , let the edges with exactly one end in C befl,f2, ..., fs, and let the edges in G\C be g, , gz, . . . , g,. Assign weight wi = 1 to each vertex i E V(C) , and wi = 0 to each vertex j e V(C). Assign weight wej = 0 to each edge ei, j = 1, . . . , r; wh = 1 to each edge fi, j = 1,. . . ,s; wgj = 2 to each edge gj, j = 1, . . . , t. Then the weight assignment is a feasible threshold assignment and the total weight is W(G) = s + 2t < r + s + t = E(G), i.e., G is not heavy. QED.

A graph G is called a split graph if the vertex set of G can be partitioned into two sets V, and Vz such that V, induces a clique and Vz induces a stable set of G.

Corollary 6.2. Split graphs with at least one edge are not heavy.

hf. G has a clique C such that E(G\C) = 0. QED.

The results in Theorem 6.1 and 5.2 suggests another characterization of heavy graphs. Although Theorem 5.2 supports the following conjecture, we have not been able to prove it.

Conjecture. A graph G is heavy if and only if for each clique C of G,

Theorem 5.2 on the triangle-free graphs says that the conjecture is true if G has no clique of size larger than 2. Also, the augment construction in Section 4 preserves inequality (9), lending further support to the conjecture.


ACKNOWLEDGMENT

The authors are grateful for the suggestions and comments of the referees. In particular, one of the referees suggested Corollary 4.7, supplied the above shorter proof of Lemma 5.1, and pointed out an omission in our origi- nal proof of Theorem 5.2. A number of the references were provided by the referees.

The authors gratefully acknowledge the support of the Air Force Office of Scientific Research under Grant AFOSR-89-0066.

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the threshold weight of a graph

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