hyperbolicity and other tree-likeness...

Hyperbolicity Easily Related Metric Graph Theory Concepts Chordality Tree-length

Hyperbolicity and Other Tree-likenessParameters

Yaokun Wu (Ç�n)- [email protected]

Department of Mathematics, Shanghai Jiao Tong University

USTC Workshop on Graph Theory and CombinatoricsMay 30, 2010


Figure: Somewhere in FeiDong with those beautiful trees in ouryounger days...


Outline

1 HyperbolicityBasic DefinitionRooted versus Unrooted0-Hyperbolic Graphs and Beyond

2 Easily Related Metric Graph Theory ConceptsEccentricityCartesian ProductBreadth Property

3 Chordality

4 Tree-length


Basic Definition

Graph Metric Space

We only consider simple, connected, unweighted, but notnecessarily finite graphs.

Any graph G together with the usual shortest-path metric on it,dG : V (G)× V (G) 7→ {0, 1, 2, . . .}, gives rise to a metric space.

We often use the shorthand xy for dG(x , y). Note that a pair ofvertices x and y form an edge if and only if xy = 1.


Basic Definition

A Function on Quartets

For any vertices x , y , u, v of a graph G, put δG(x , y , u, v) to bethe difference between the largest and the second largest ofthe following three quantities

xu + yv2

,xv + yu

2, and

xy + uv2

.


Basic Definition

δ(x , x , u, v) = 0

If x = y , then we have

xu + yv= xv + yu= xu + xv ≥ uv =xy + uv .

By symmetry, this means that δ(x , y , u, v) = 0 providedx , y , u, v are not 4 different vertices.


Basic Definition

Tree and the Four-Point Condition (4PC)

ry

rur

v

rx ra rb

xu + yv = xv + yu = xy + uv + 2ab ⇒ δ(x , y , u, v) = 0


Basic Definition

Hyperbolicity

A graph G, viewed as a metric space, is δ-hyperbolicprovided for any vertices x , y , u, v in G it holdsδ(x , y , u, v) ≤ δ

and the (Gromov) hyperbolicity of G,denoted δ∗(G), is the minimum half integer δ such that G isδ-hyperbolic.

Note that it may happen δ∗(G) = ∞. But for a finite graphG, δ∗(G) is clearly finite and polynomial time computable.


Basic Definition

Hyperbolicity

A graph G, viewed as a metric space, is δ-hyperbolicprovided for any vertices x , y , u, v in G it holdsδ(x , y , u, v) ≤ δ and the (Gromov) hyperbolicity of G,denoted δ∗(G), is the minimum half integer δ such that G isδ-hyperbolic.

Note that it may happen δ∗(G) = ∞. But for a finite graphG, δ∗(G) is clearly finite and polynomial time computable.


Basic Definition

Hyperbolicity

A graph G, viewed as a metric space, is δ-hyperbolicprovided for any vertices x , y , u, v in G it holdsδ(x , y , u, v) ≤ δ and the (Gromov) hyperbolicity of G,denoted δ∗(G), is the minimum half integer δ such that G isδ-hyperbolic.Note that it may happen δ∗(G) = ∞. But for a finite graphG, δ∗(G) is clearly finite and polynomial time computable.


Basic Definition

Tree Metric

Every tree has hyperbolicity zero.

A metric space is a real tree (Tits 1977) if and only if it ispath-connected and 0-hyperbolic.

The hyperbolicity of a graph is a way to measure the additivedistortion with which every four-points sub-metric of the givengraph metric embeds into a tree metric, namely it is atree-likeness parameter.


Basic Definition

Weakly Geodesic Graphs

A weakly geodesic graph is a graph in which any two vertices atdistance two have exactly one common neighbor.

If G is not weakly geodesic, we can find x , y , u, v such thatxy = 2 and u, v are two different vertices lying in the geodesicsbetween x and y .This gives

xy + uv ≥ 3 > 2 = xu + yv = xv + yu

and hence δ∗(G) ≥ δ(x , y , u, v) ≥ 12 .

We now see that every 0-hyperbolic graph is weakly geodesic.


Basic Definition



If G is not weakly geodesic, we can find x , y , u, v such thatxy = 2 and u, v are two different vertices lying in the geodesicsbetween x and y .

This gives

xy + uv ≥ 3 > 2 = xu + yv = xv + yu




Basic Definition



If G is not weakly geodesic, we can find x , y , u, v such thatxy = 2 and u, v are two different vertices lying in the geodesicsbetween x and y .This gives

xy + uv ≥ 3 > 2 = xu + yv = xv + yu




Basic Definition

Cycle

The hyperbolicity of the n-cycle Cn is bn4c −

12 if n is congruent

to 1 modulo 4 and is bn4c else.

In particular, Cn has nonzero hyperbolicity for n > 3. Note thatCn is weakly geodesic when n > 4.


Basic Definition

Mikhail Gromov

The concept of hyperbolicity comes from the work of Gromov ingeometric group theory which encapsulates many of the globalfeatures of the geometry of complete, simply connectedmanifolds of negative curvature. – M. Bridson, A. Haefliger,Metric Spaces of Non-Positive Curvature, Springer, 1999.

It is incredible what Mikhail Gromov can do, just with thetriangle inequality. – Dennis Sullivan


Rooted versus Unrooted

Gromov Product

For any vertex a ∈ V (G), the Gromov product, also known asthe overlap function, of any two vertices x and y of G withrespect to a is equal to 1

2(xa + ya− xy) and is denoted by(x · y)a.

We say that G is δ-hyperbolic with respect to a ∈ V (G) if for anyx , y , v ∈ G it holds

(x · y)a ≥ min((x · v)a, (y · v)a)− δ.

It is easy to check that G is δ-hyperbolic if and only if G isδ-hyperbolic with respect to every vertex of G.


Rooted versus Unrooted

Farris Transform in Mathematics and Phylogenetics

Think of a as the ancestor of all species, take a very largenumber ρ, then ρ− (x · y)a is an estimate for the number ofyears that have passed since the last common ancestor of xand y existed.

4 A. Dress, K.T. Huber, and V. Moulton

a

xy

lca(x,y)

Figure 2: The dotted lines depict D(a,ρ)(x,y) which can be thought of as an estimate forthe number of years that have passed since lca(x,y) existed.

in view of

ρ+12

(DGC(x, y)−DGC(x, a)−DGC(y, a)) = ρ−DGC (a, lca(x, y)) =

ρ−DGC(a, v)−DGC(lca(x, y), v) = ρ0−D∗GC(lca(x, y), v) =

D∗GC(z, v)−D∗

GC(lca(x, y), v) = D∗GC (z, lca(x, y))

for both, z := x and z := y.Considerations like this may have led Farris and his collaborators in 1970 to de-

fine, for any dissimilarity D : X ×X → R : (x, y) 7→ xy := D(x, y), any element a ∈ Xrepresenting a putative outgroup, and any large positive real number ρ, the map

D(a,ρ) : X ×X → R,

later to be dubbed the Farris transform of D relative to a and ρ, by

D(a,ρ)(x, x) := 0,

for all x ∈ X , and by

D(a,ρ)(x, y) := ρ+12(xy− xa− ya),

for all x, y ∈ X with x 6= y, and to suggest these numbers D(a,ρ)(x, y) as likely estimates,up to a fixed proportionality factor, for the number of years that passed since lca(x, y)existed (cf. Figure 2).

This transformation, originally suggested by Farris et al. in [34, p. 181] (see also[32, p. 83, 5ff] and [33, p. 491]) was dubbed the Farris transform in [4] (see also [6]),and it was rediscovered in the context of phylogenetics in both, [11] and [41]. Intrigu-ingly, the same transformation has also come up in investigations of apparently ratherunrelated mathematical topics: For example, it was implicitly used in the study of hy-perbolic groups by Gromov [36, p. 158] (see also [10, p. 134]) and later also dubbed,in [17], the Gromov product. In disguise, it appears also in Distance Geometry where itis known as the covariance mapping (see [23, p. 56] for more details).

The Farris transform based at a sends the given (graph) metricdG to the map

Dρ,a(dG) : V (G)× V (G) → R : (x , y) 7→ ρ− (x · y)a.


0-Hyperbolic Graphs and Beyond

Block Tree

A maximal connected subgraph without a cut vertex is called ablock.

The hyperbolicity of a graph is the maximum hyperbolicityof its blocks.

rg

re r f

r d

r c

ra rb-

rg

re rfrdrdrcrc

ra rb

B3

B2

B1

-

rB3

rdrB2

rcrB1

The block tree of G has the cut vertices and blocks of G as itsvertices and cB is an edge if c is a cut vertex inside the blockB.The block tree is always a tree.



Block Tree

A maximal connected subgraph without a cut vertex is called ablock.The hyperbolicity of a graph is the maximum hyperbolicityof its blocks.

rg

re r f

r d

r c

ra rb-

rg

re rfrdrdrcrc

ra rb

B3

B2

B1

-

rB3

rdrB2

rcrB1




Block Tree


rg

re r f

r d

r c

ra rb-

rg

re rfrdrdrcrc

ra rb

B3

B2

B1

-

rB3

rdrB2

rcrB1

The block tree of G has the cut vertices and blocks of G as itsvertices and cB is an edge if c is a cut vertex inside the blockB.

The block tree is always a tree.



Block Tree


rg

re r f

r d

r c

ra rb-

rg

re rfrdrdrcrc

ra rb

B3

B2

B1

-

rB3

rdrB2

rcrB1




Block Graphs = 0-Hyperbolic Graphs

Howorka (1979), Bandelt, Mulder (1986), Dress, Moulton, Steel(1997)

A block graph is a graph whose blocks are all cliques.

Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.

If thiscycle is an isometric cycle, we see that δ∗(G) > 0. If thiscycle is not an isometric cycle, we will find two maximalcliques of G sharing at least two vertices, which impliesthat G is not weakly geodesic and hence δ∗(G) > 0 alsofollows.





A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.

A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.






A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.

If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.






A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.






A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.If thiscycle is an isometric cycle, we see that δ∗(G) > 0.

If thiscycle is not an isometric cycle, we will find two maximalcliques of G sharing at least two vertices, which impliesthat G is not weakly geodesic and hence δ∗(G) > 0 alsofollows.





A block graph is a graph whose blocks are all cliques.Since complete graphs are 0-hyperbolic, so are blockgraphs.A graph is a block graph if and only if every cycle of thegraph induces a clique.If a graph G is not a block graph, we can find a shortestcycle in it which does not induce a complete graph.If thiscycle is an isometric cycle, we see that δ∗(G) > 0. If thiscycle is not an isometric cycle, we will find two maximalcliques of G sharing at least two vertices, which impliesthat G is not weakly geodesic and hence δ∗(G) > 0 alsofollows.



12 -Hyperbolic Graphs

Bandelt and Chepoi (2003): A graph is 12 -hyperbolic if and

only if it does not contain a set of six special graphs asisometric subgraphs and all balls of the graph are convex.

Bandelt and Chepoi (2008): “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.



12 -Hyperbolic Graphs

Bandelt and Chepoi (2003): A graph is 12 -hyperbolic if and

only if it does not contain a set of six special graphs asisometric subgraphs and all balls of the graph are convex.Bandelt and Chepoi (2008): “a characterization of all1-hyperbolic graphs by forbidden isometric subgraphs isnot in sight, in as much as isometric cycles of lengths up to7 may occur, thus complicating the picture”.



Structure Tree Decomposition

The block tree is a decomposition of 1-connected graphsinto 2-connected blocks.

Tutte defined a tree decomposition of 2-connected graphsinto 3-connected blocks.Dunwoody and Krön (2010) define a structure tree thatcontains information about k -connectivity of a graph forany k .Can we relate the behavior of the hyperbolicity parameterto the structure tree decomposition?




The block tree is a decomposition of 1-connected graphsinto 2-connected blocks.Tutte defined a tree decomposition of 2-connected graphsinto 3-connected blocks.

Dunwoody and Krön (2010) define a structure tree thatcontains information about k -connectivity of a graph forany k .Can we relate the behavior of the hyperbolicity parameterto the structure tree decomposition?




The block tree is a decomposition of 1-connected graphsinto 2-connected blocks.Tutte defined a tree decomposition of 2-connected graphsinto 3-connected blocks.Dunwoody and Krön (2010) define a structure tree thatcontains information about k -connectivity of a graph forany k .

Can we relate the behavior of the hyperbolicity parameterto the structure tree decomposition?




The block tree is a decomposition of 1-connected graphsinto 2-connected blocks.Tutte defined a tree decomposition of 2-connected graphsinto 3-connected blocks.Dunwoody and Krön (2010) define a structure tree thatcontains information about k -connectivity of a graph forany k .Can we relate the behavior of the hyperbolicity parameterto the structure tree decomposition?


Eccentricity

Eccentricity

For any u ∈ V (G),

ecc(u) = maxv∈V (G)

uv ,

rad(G) = minu∈V (G)

ecc(u),

diam(G) = maxu∈V (G)

ecc(u).

The center of G, denoted C(G), is

{u ∈ V (G) : ecc(u) = rad(G)}.


Eccentricity

Center

Camille Jordan (1869): Let G be a tree. Then xy ≤ 1 forany x , y ∈ C(G).

The center of a 12 -hyperbolic graph is convex.

Chepoi, Dragon, Estellon, Habib, Vaxés, Xiang (2008): LetG be a graph. Then xy ≤ 4δ∗(G) + 1 for any x , y ∈ C(G).

Proof.For any x , y ∈ C(G), take vertices v and u such thatmax{xv , yv} ≤ d xy

2 e and uv = ecc(v).Note that uv ≥ rad(G) ≥ max{xu, yu}.It follows thatxy ≤ max{xu + yv − uv , xv + yu − uv}+ 2δ∗ ≤max{xv , yv}+ 2δ∗(G) ≤ dxy

2 e+ 2δ∗(G), implying thatxy ≤ 4δ∗(G) + 1.


Eccentricity

Center

Camille Jordan (1869): Let G be a tree. Then xy ≤ 1 forany x , y ∈ C(G).The center of a 1

2 -hyperbolic graph is convex.

Chepoi, Dragon, Estellon, Habib, Vaxés, Xiang (2008): LetG be a graph. Then xy ≤ 4δ∗(G) + 1 for any x , y ∈ C(G).





Eccentricity

Center


2 -hyperbolic graph is convex.Chepoi, Dragon, Estellon, Habib, Vaxés, Xiang (2008): LetG be a graph. Then xy ≤ 4δ∗(G) + 1 for any x , y ∈ C(G).





Eccentricity

Center


2 -hyperbolic graph is convex.Chepoi, Dragon, Estellon, Habib, Vaxés, Xiang (2008): LetG be a graph. Then xy ≤ 4δ∗(G) + 1 for any x , y ∈ C(G).


2 e and uv = ecc(v).Note that uv ≥ rad(G) ≥ max{xu, yu}.It follows thatxy ≤ max{xu + yv − uv , xv + yu − uv}+ 2δ∗ ≤max{xv , yv}+ 2δ∗(G) ≤ d xy



Eccentricity

An Example from Gerard Chang (1991)

c s sc c csx sys

us

v

Figure: A chordal graph G whose center is marked with bullets.

(xv + yu, xu + yv , xy + uv) = (5, 4, 1), δ∗(G) = δ(x , y , u, v) = 12 .

The diameter of C(G) is 3 = 4× 12 + 1.


Eccentricity

Hyperbolicity and Small World

The hyperbolicity of a graph with diameter D is at mostbD

2 c.Proof

Many experiments say that some large practicalcommunication networks have surprisingly lowhyperbolicity (and so simple enough to deserve somemathematical study?).


Cartesian Product

WU, Zhang (Ü¤+) 2009: Let G1 and G2 be two graphs ofhyperbolicity 0. Then δ∗(G1�G2) = min(D1, D2), where D1 andD2 are the diameters of G1 and G2, respectively.

ry

ru

rvrxrr rr

r rrrδ∗ = δ(x , y , u, v) =

10− 62

= 2


Breadth Property

For any two vertices x and y of a graph G, set brG(x , y) tobe max{uv : xu = xv = xy − uy = xy − vy}. The breadthof G, denoted br(G), is maxx ,y∈V (G) brG(x , y).

It is clear that br(G) = 0 when G is a tree.In general, given xu = xv = xy − yu = xy − yv , we havexu + yv = xv + yu = xy = (xy + uv)− uv .This says that br(G) ≤ 2δ∗(G).


Breadth Property

For any two vertices x and y of a graph G, set brG(x , y) tobe max{uv : xu = xv = xy − uy = xy − vy}. The breadthof G, denoted br(G), is maxx ,y∈V (G) brG(x , y).It is clear that br(G) = 0 when G is a tree.

In general, given xu = xv = xy − yu = xy − yv , we havexu + yv = xv + yu = xy = (xy + uv)− uv .This says that br(G) ≤ 2δ∗(G).


Breadth Property

For any two vertices x and y of a graph G, set brG(x , y) tobe max{uv : xu = xv = xy − uy = xy − vy}. The breadthof G, denoted br(G), is maxx ,y∈V (G) brG(x , y).It is clear that br(G) = 0 when G is a tree.In general, given xu = xv = xy − yu = xy − yv , we havexu + yv = xv + yu = xy = (xy + uv)− uv .

This says that br(G) ≤ 2δ∗(G).


Breadth Property

For any two vertices x and y of a graph G, set brG(x , y) tobe max{uv : xu = xv = xy − uy = xy − vy}. The breadthof G, denoted br(G), is maxx ,y∈V (G) brG(x , y).It is clear that br(G) = 0 when G is a tree.In general, given xu = xv = xy − yu = xy − yv , we havexu + yv = xv + yu = xy = (xy + uv)− uv .This says that br(G) ≤ 2δ∗(G).


Length of the Longest Chordless Cycle

A graph is k -chordal if it does not contain any induced n-cyclefor any n > k . The chordality of G, denoted lc(G), is theminimum integer k ≥ 2 such that G is k -chordal.

A 3-chordal graph is usually just called a chordal graph.An especially rich theory on chordal graphs has beendeveloped in an astonishingly wide area of mathematics andstatistics and other applied fields.




A 3-chordal graph is usually just called a chordal graph.

An especially rich theory on chordal graphs has beendeveloped in an astonishingly wide area of mathematics andstatistics and other applied fields.




A 3-chordal graph is usually just called a chordal graph.An especially rich theory on chordal graphs has beendeveloped in an astonishingly wide area of mathematics andstatistics and other applied fields.


Low Hyperbolicity ; Low Chordality

Indeed, take any graph G and form the new graph G′ by addingan additional vertex and connecting this new vertex with everyvertex of G. It is obvious that δ∗(G′) ≤ 1 while lc(G′) = lc(G) ifG is not a tree. Moreover, it is equally easy to see that G′ iseven 1

2 -hyperbolic if G does not have any induced 4-cycle(Koolen, Moulton, 2002)

Surely, this example does not preclude the possibility that formany important graph classes we can bound their chordalityfrom above in terms of their hyperbolicity.


BKM Theorem

Theorem 1 (Brinkmann, Koolen, Moulton, 2001)

Every chordal graph is 1-hyperbolic and it is 12 -hyperbolic if and

only if it contains neither H1 nor H2 as an isometric subgraph.

ry

ru r v

rxra r b

rc r d

H1

ry

ru rvrxra rb

rc rdH2


Low Chordality ⇒ Low Hyperbolicity

Theorem 2 (WU, Zhang, 2009)

For each k ≥ 4, all k-chordal graphs are b k2 c2 -hyperbolic.


Tightness of Theorem 2

It is clear that if the bound claimed by Theorem 2 is tight fork = 4t (k = 4t − 2) then it is tight for k = 4t + 1 (k = 4t − 1).Consequently, Examples 3 and 4 to be presented below indeedmean that the bound reported in Theorem 2 is tight for everyk ≥ 4.


Example 3

The chordality of the k -cycle is surely k . Also recall that

δ∗(Ck ) =

b k

2 c2 , if k ≡ 0 (mod 4);

b k2 c2 + 1

2 , else.

Example 4

For any t ≥ 2 we set Ft to be the outerplanar graph obtainedfrom the 4t-cycle [v1v2 · · · v4t ] by adding the two edges {v1, v3}and {v2t+1, v2t+3}. Clearly, δ(v2, vt+2, v2t+2, v3t+2) = t − 1

2 .Furthermore, we can check that lc(Ft) = 4t − 2 and

δ∗(Ft) = t − 12 = δ(v2, vt+2, v2t+2, v3t+2) = lc(Ft )

4 =b lc(Ft )

2 c2 .


xy + uv = 3 + 4 = 7, xu + yv = xv + yu = 2 + 2 = 4 ⇒δ(x , y , u, v) = 3

2 .

rxra

ru

rc

rb rvrd

ry

Figure: The graph F2 has hyperbolicity 32 , tree-length 2 and

chordality 6.


Let C4, H1, H2, H3, H4 and H5 be the graphs displayed in nexttwo slides. It is simple to check that each of them hashyperbolicity 1 and is 5-chordal.


ryru

rvrx

C4

ry

r

u rvrxra rb

rc rdH1

ry

ru rvrxra rb

rc rdH2


ry

ru rvrxrc rd

H3

ry

ru rvrxra rb

rc rdH4

ry

rc

ra rbrdru rv

rx

H5

Figure: Six 5-chordal graphs with hyperbolicity 1.


Theorem 5 (WU, Zhang, 2009)A 5-chordal graph has hyperbolicity one if and only if one ofC4, H1, H2, H3, H4, H5 appears as an isometric subgraph of it.

Together with Theorem 2, this generalizes the BKM Theorem(Theorem 1).


rr

r

r r

r rrr

G1

rr r

r r

r rr

rG2

rr

rr

rr

rG3

rr r

rr r

C6

Figure: Four graphs with hyperbolicity 1 and chordality 6.


Conjecture 6 (WU, Zhang, 2009)

A 6-chordal graph is 12 -hyperbolic if and only if it does not

contain any of a list of ten special graphsG1, G2, G3, C6, C4, Hi , i = 1, . . . , 5, as an isometric subgraph.


Tree-length

The tree-length of a graph G, denoted tl(G), wasintroduced by Dourisboure and Gavoille in 2007 and is theminimum integer k such that there is a chordal graph G′

satisfying V (G) = V (G′), E(G) ⊆ E(G′) andmax(dG(u, v) : dG′(u, v) = 1) = k . We use the conventionthat the tree-length of the graph with one vertex is 1.

It is straightforward from the definition that chordal graphsare exactly the graphs of tree-length 1.


Tree-length

The tree-length of a graph G, denoted tl(G), wasintroduced by Dourisboure and Gavoille in 2007 and is theminimum integer k such that there is a chordal graph G′

satisfying V (G) = V (G′), E(G) ⊆ E(G′) andmax(dG(u, v) : dG′(u, v) = 1) = k . We use the conventionthat the tree-length of the graph with one vertex is 1.It is straightforward from the definition that chordal graphsare exactly the graphs of tree-length 1.


Tree Decomposition

The concept of tree decompositions was introduced byRobertson and Seymour in 1984 and has since beenextensively studied in both mathematics and lots of appliedfields. It specify a very nice way to view a graph as a tree.

A tree decomposition (T , S) of a graph G is a tree T such thateach vertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:

(Vertex Covering) ∪v∈V (T )Sv = V (G).(Edge Covering) For any edge {u, w} ∈ E(G) there existsv ∈ V (T ) such that u, w ∈ Sv .

(Running Intersection Property) For any u ∈ V (G),{v ∈ V (T ) : u ∈ Sv} induces a subtree of T . In otherwords, for any v , w ∈ V (T ), Sv ∩ Sw can be seen in everybag along the path connecting v and w in T .


Tree Decomposition

The concept of tree decompositions was introduced byRobertson and Seymour in 1984 and has since beenextensively studied in both mathematics and lots of appliedfields. It specify a very nice way to view a graph as a tree.A tree decomposition (T , S) of a graph G is a tree T such thateach vertex v of T corresponds to a bag Sv ⊆ V (G) and thefollowing conditions are satisfied:




Tree Decomposition


(Vertex Covering) ∪v∈V (T )Sv = V (G).

(Edge Covering) For any edge {u, w} ∈ E(G) there existsv ∈ V (T ) such that u, w ∈ Sv .



Tree Decomposition





Tree-length is a Tree-likeness Parameter

The length of a tree decomposition of a graph G is themaximum distance in G between two vertices in the same bagof the decomposition. The length is a measure of the likenessof G to the tree T according to the given tree decomposition(T , S).

The tree-length of a graph G, tl(G), turns out to be the shortestlength of all tree decompositions of G.

Example 7 (Dourisboure and Gavoille (2007))

The tree-length of an n-cycle is dn3e.


Finite Characterization: Tree-length vs Hyperbolicity

From the tree decomposition definition of tree-length, we caneasily see that tl(H) ≤ tl(G) if H is a minor of G. By the GraphMinors Theorem of Robertson and Seymour, for any given k ,there exists a finite excluded minor characterization for thosegraphs whose tree length is at most k .

Hyperbolicity does not have the minor-monotone property.Instead, for every isometric subgraph H of a given graph G, wehave δ∗(H) ≤ δ∗(G). But, even 0-hyperbolic graphs may nothave a finite excluded isometric subgraph characterization.


Tree-length and Hyperbolicity are Comparable

The following are two results from Chepoi, Dragan, Estellon,Habib, Vaxés (2008).

Theorem 8The inequality tl(G) ≤ 12k + 8k log2 n + 17 holds for anyk-hyperbolic graph G with n vertices.

Theorem 9A graph G is k-hyperbolic provided its tree-length is no greaterthan k .

Proof of Theorem 9

Since chordal graphs have tree-length 1, the first part of theBKM Theorem (Theorem 1) directly follows from Theorem 9.


Tightness of Theorem 9: δ∗(G2k+1,2k+1) = tl(G2k+1,2k+1)

Example 10

For any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1.

Dourisboure and Gavoile (2007)showed that the tree-length of Gn,m is min(n, m) if n 6= m orn = m is even and is n − 1 if n = m is odd. Our result onCartesian product shows that δ∗(Gm,n) = min(m, n)− 1. Thissays that Theorem 9 is tight.



Example 10

For any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile (2007)showed that the tree-length of Gn,m is min(n, m) if n 6= m orn = m is even and is n − 1 if n = m is odd.

Our result onCartesian product shows that δ∗(Gm,n) = min(m, n)− 1. Thissays that Theorem 9 is tight.



Example 10

For any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile (2007)showed that the tree-length of Gn,m is min(n, m) if n 6= m orn = m is even and is n − 1 if n = m is odd. Our result onCartesian product shows that δ∗(Gm,n) = min(m, n)− 1.

Thissays that Theorem 9 is tight.



Example 10

For any two natural numbers m and n, the grid graph Gm,n isthe graph with vertex set {1, 2, . . . , m} × {1, 2, . . . , n} and (i1, i2)and (j1, j2) are adjacent in Gm,n if any only if(i1 − j1)2 + (j1 − j2)2 = 1. Dourisboure and Gavoile (2007)showed that the tree-length of Gn,m is min(n, m) if n 6= m orn = m is even and is n − 1 if n = m is odd. Our result onCartesian product shows that δ∗(Gm,n) = min(m, n)− 1. Thissays that Theorem 9 is tight.


Chordality and Tree-length

What follows is a result of Gavoille, Katz, Katz, Paul, Peleg(2001).

Theorem 11

If G is a k-chordal graph, then tl(G) ≤ bk2c.

Proof.To obtain a minimal triangulation of G, it suffices to select amaximal set of pairwise parallel minimal separators of G andadd edges to make each of them a clique (Parra, Scheffler,1997). It is easy to check that each such new edge connectstwo points of distance at most b k

2c apart in G.




Theorem 11

If G is a k-chordal graph, then tl(G) ≤ b k2c.


2c apart in G.




Theorem 11


Proof.To obtain a minimal triangulation of G, it suffices to select amaximal set of pairwise parallel minimal separators of G andadd edges to make each of them a clique (Parra, Scheffler,1997).

It is easy to check that each such new edge connectstwo points of distance at most b k

2c apart in G.




Theorem 11



2c apart in G.


Proof of Theorem 11

●

●

●

●

u

●

v

●

●

●

●

●

C B

●●

●

● ●●

●

|C|+ |B| = 4 + 6 ≤ k ⇒ uv ≤ |C| = 4 ≤ bk2c


An Easy proof of a Weaker Result

The following result is weaker than our Theorem 2. It wasnotified to us by Dragan and is presumably in the folklore.

Theorem 12

Every k-chordal graph is b k2c-hyperbolic.

Proof.

By Theorem 11, lc(G) ≤ k ⇒ tl(G) ≤ bk2c;

By Theorem 9, tl(G) ≤ bk2c ⇒ δ∗(G) ≤ bk

2c.


A Chordal Graph Sandwich Problem

Dourisboure and Gavoille (2007) posed the open problemthat whether or not

tl(G) ≤ d lc(G)

3e (1)

is true. They already knew that for any outerplanar graphG, tl(G) = d lc(G)

3 e holds.

The kth-power of a graph G, denoted Gk , is the graph withV (G) as vertex set and there is an edge connecting twovertices u and v if and only if dG(u, v) ≤ k .

Problem reformulation: For any graph G, is there always achordal graph H such that V (H) = V (G) = V (Gd lc(G)

3 e) andE(G) ⊆ E(H) ⊆ Gd lc(G)

3 e?


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Appendix

Proofδ(x , y , u, v) ≤ bD

2 c

Assume xy + uv ≥ xu + yv ≥ xv + yuand hence δ = (xy+uv)−(xu+yv)

2 .

By the triangle inequality,

xu + yu ≥ xy , ux + vx ≥ uv ,

xv + yv ≥ xy , vy + uy ≥ uv .

Summing up the above yields

(xu + yv) + (xv + yu) ≥ xy + uv .

This gives δ ≤ xy+uv4 ≤ D

2 .

When D is odd, we show thatδ = D

2 is impossible.

Otherwise, we havexv + yu = D, xu + yu = xy =D, ux + vx = uv = D, whichsays that 3D is an evennumber, a contradiction. Return

Appendix

Proof of Theorem 9

ry ∈ Y

ru ∈ U

rv ∈ V

rx ∈ X

rA rB

Let Q = {xu + yv , xv +yu, xy + uv}, α =xB + yB + uB + vB.

maxQ ≤ α + 2diam(B)

The largest two elementsof Q is at least α.

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