CENTERS OF CHORDAL GRAPHS

Graphs and Combinatorics 7, 305-313 (1991)

Graphs and

Combinatorics

© Springer-Verlag 1991

Centers of Chordal Graphs*

G e r a r d J. C h a n g

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China

Abstract. In a graph G = (V, E), the eccentricity e(S) of a subset S _~ V is m a x ~ v mint ~ s d(x, y); and e(x) stands for e({x}). The diameter of G is maxx~ v e(x), the radius r(G) of G is mine v e(x) and the clique radius cr(G) is mine(K) where K runs over all cliques. The center of G is the subgraph induced by C(G), the set of all vertices x with e(x) = r(G). A clique center is a clique K with e(K) = cr(G). In this paper, we study the problem of determining the centers of chordal graphs. It is shown that the center of a connected chordal graph is distance invariant, biconnected and of diameter no more than 5. We also prove that 2cr(G)<_ d(G)< 2cr(G)+ 1 for any connected chordal graph G. This result implies a characterization of a biconnected chordal graph of diameter 2 and radius 1 to be the center of some chordal graph.

1. Introduction

I n a g r a p h G = (V, E), the distance d(x, y) from vertex x to vertex y is the m i n i m u m n u m b e r o f edges in a p a t h f r o m x to y. T h e eccentricity e(x) of a vertex x is the m a x i m u m d i s t a n c e f r o m x to a n y vertex in G. T h e diameter d(G) of G is the m a x i m u m e c c e n t r i c i t y o f a vertex in G a n d radius r(G) the m i n i m u m eccentricity. D e n o t e b y C(G) the set of all vertices whose eccentricities a r e e q u a l to r(G). T h e center of G is the s u b g r a p h ( C ( G ) ) i n d u c e d by C(G).

It was s h o w n in [7] t h a t the c e n t e r o f a g r a p h lies w i t h i n a single b l o c k ( b i c o n n e c t e d c o m p o n e n t ) , b u t need n o t be a block. As d e s c r i b e d in [ l ] , H e d e t n i e m i p r o v e d t h a t a n y g r a p h H is i s o m o r p h i c to the c e n t e r o f s o m e g r a p h G which is of d i a m e t e r 4 a n d r a d i u s 2. In fact, G can be o b t a i n e d from H b y a d d i n g four new vertices u, v, w, x such t h a t v a n d w a r e a d j a c e n t to all vertices o f H, u is a d j a c e n t o n l y to v a n d x o n l y to w. H o w e v e r , the centers o f s o m e special g r a p h s are restricted. T h e o l d e s t result is J o r d a n ' s w e l l - k n o w n t h e o r e m for trees [8]: the c e n t e r of a tree is e i t h e r K I o r K z . As a n e a s y g e n e r a l i z a t i o n we can s a y t h a t the c e n t e r of a c o n n e c t e d b l o c k g r a p h , i.e. a g r a p h w h o s e b l o c k s are c o m p l e t e g r a p h s , is e i t h e r a c u t - v e r t e x o r a block. P r o s k u r o w s k i [10] p r o v e d t h a t the c e n t e r of a m a x i m a l

* Supported by the National Science Council of the Republic of China under grant NSC77-0208-M008-05

outplanar graph is one of seven special graphs. As a generalization, in [11] he found all possible centers of 2-trees, and showed that the center of a 2-tree is biconnected.

A graph is chordal (triangulated or rigid circuit) if every cycle of length greater than three possesses a chord, i.e. an edge joining two nonconsecutive vertices of the cycle. Chordal graphs were first introduced by Hajnal and Sur~nyi [6] and then studied extensively by many people, see [5] for general results. The class of chordal graphs contains trees, block graphs, maximal outerplanar graphs and 2-trees. It was shown in [9] that the center of a chordal graph is connected. The main purpose of this paper is to study the centers of chordal graphs and to answer a part of the question given by Duchet [4]: determine the centers of chordal graphs.

Section 2 introduces the idea of clique radius cr(G) of a graph G, and proves a main theorem: 2cr(G) < d(G) < 2cr(G) + 1 for any connected chordal graph G. This result is used in Section 3 as the key for a characterization of centers of some chordal graphs.

Section 3 studies necessary and sufficient conditions for the centers of chordal graphs. In particular, we prove that the center of a chordal graph is distance invariant, biconnected and of diameter no more than 5. Finally, by using the main theorem in Section 2, we give a necessary and sufficient condition for a biconnected chordal graph of diameter 2 and radius 1 to be the center of some chordal graph.

2. Clique Centers of Chordal Graphs

A clique of a graph is a set of pairwise adjacent vertices. In a graph G = (1/, E), the distance d(x, S) from a vertex x to a set S _ F is miny~sd(x, y). The eccentricity e(S) of a set S of vertices is the maximum distance from any vertex to S. A clique center of G is a clique with minimum eccentricity which is called the clique radius of G and is denoted by cr(G). This idea is similar to bi-center which is an edge with minimum eccentricity; see Theorem 4.2 in [3].

The main result of this section is the relation between clique radius and diame- ter of a connected chordal graph. It is the keystone for determining the necessary and sufficient conditions of a chordal graph of diameter 2 and radius 1 to be the center of some chordal graph.

Theorem

2.1. 2cr(G) < d(G) <_ 2cr(G) + 1 for any connected chordal graph G =

(r,F~).

Before proving Theorem 2.1, we first list some definitions and results from [2] which are needed in this paper. Lemma 2.4 is from [9].

If d(x,y) = k is finite and 0 < m _< k, then Bet(x, m, y) denotes the set of all vertices z between x and y such that d(x, z) = m and d(z, y) = k - m. An n-sun is a chordal graph of 2n vertices with a Hamiltonian cycle (yl, zl, Y2, 2"2,..., Yn, Zn, Ya) and each Yi is of degree two. Equivalently, an n-sun is a chordal graph G = (V, E) whose vertex set V can be partitioned into Y = (Yl,-..,Y,} and Z = {z 1 .. . . . z,} such that the following three conditions hold.

(S1) Y is a stable set in G. (S2) (z 1 .. . . . z , , z l ) is a cycle in G.

Centers of Chordal Graphs 307 ($3) (Yi, zi) e E if and only if i --- j or i = j + 1 (mod n).

In the a b o v e definition, if Z is a clique, then we call the n-sun a complete n-sun. L e m m a 2.2. I f C is a cycle in a chordal graph, then for every edge (u, v) of C there is a vertex w of C which is adjacent to both u and v.

L e m m a 2.3. [2] I f G is chordal and d(x, y) = k, then Bet(x, m, y) is a clique for any O < m < _ k .

L e m m a 2.4. [9] In a chordal graph G, if K is a clique and x a vertex such that d(x, y) = k is a constant for all y e K, then Bet(z, 1, x) and B e t ( w , 1, x) are comparable for any z, w E K (i.e. one is a subset of the other); consequently, (']y~K Bet(y, 1,x) is

not empty.

Theorem 2.5. [2] 2r(G) - 2 < d(G) < 2r(G) for any connected chordal graph G.

Moreover, if 2r(G) - 2 = d(G), then G has a 3-sun as an induced subgraph. W e can also p r o v e two slightly m o r e general results as follows.

T h e o r e m 2.6. I f X and Y are two cliques in a chordal graph G such that d(x, y) = k is a constant for all x ~ X and y E Y, then Bet(X, m, Y) =- U {Bet(x, m, y): x e X and y ~ Y} is a clique for any 0 < m < k.

Proof. Consider the g r a p h G* obtained from G by adding two new vertices u and v which are adjacent to all vertices in X a n d Y respectively. T h e n G* is a chordal g r a p h a n d d(u, v) = k + 2. The t h e o r e m follows from L e m m a 2.3 and the fact that

Bet(X, m, Y) = Bet(u, m + 1, v). []

Theorem 2.7. In a chordal graph G, if K is a clique and x is a vertex such that

d(x, y) = k is a constant for all y ~ K, then Bet(z, m, x) and Bet(w, m, x) are compara- ble for any 1 ~ m <_ k and z, w ~ K; consequently, (~y~KBet(y,m,x) is not empty. Proof. The t h e o r e m is true for m = 1 by L e m m a 2.4. Suppose it is true for m - 1. Let Z = Bet(z,m - 1,x) and W = B e t ( w , m - 1,x). Z ~_ W or W _ Z by the in- duction hypothesis. T h e o r e m 2.6 implies that ZI3 W is a clique. By L e m m a 2.4, Bet(y, 1,x)'s are c o m p a r a b l e for all y ~ Z U W. Bet(z,m,x) = U r ~ z Bet(y, 1,x) and B e t ( w , m , x ) = U y ~ w B e t ( y , 1,x) then imply that Bet(z,m,x) and B e t ( w , m , x ) are

c o m p a r a b l e . [ ]

T h e g r a p h in Fig. 1 shows that Bet(x 1, 1,yl) and Bet(xz, 1,y2) are not c o m p a - rable, so we c a n n o t get a generalization of T h e o r e m 2.7 or L e m m a 2.4 by replacing vertex x by a cliue.

N o w w e are r e a d y to p r o v e the m a i n t h e o r e m of this section.

P r o o f of Theorem 2.1. C h o o s e a clique center K such that S(K) = {x e V: d(x, K) = cr(G)} has smallest n u m b e r of vertices. S u p p o s e x, y e V are such that d(x, y) = d(G). C h o o s e x*, y* e K with d(x, x*) = d(x, K) and d(y, y*) = d(y, K). Then

zl ~

Yl

x2

Y2

Fig. 1. Bet(x 1, 1, Y a) and Bet(x2, 1, Y2) are not comparable

Suppose that d(G) < 2cr(G) - 1. C h o o s e a fixed vertex w e S(K). Let K* = {w* e K: d(w,w*) = cr(G)}. Suppose K = K*, i.e. d(w,w*) = cr(G) for all w* e K. By L e m m a 2.4, there is a vertex x e (~w.~KBet(w*, 1,w). T h e n K U x is a clique center with S(K U x) c S(K) - w, a contradiction to the minimality of IS(K)L. So K* is a p r o p e r subset of K.

Next, we consider the set T = {x e V: cr(G) - 1 < d(x, K) < d(x, K*)}. If T = 2~, then K* is a clique center with S(K*) = S(K). T h e same a r g u m e n t s in the second p a r a g r a p h lead to a contradiction. So T ~ ~. F o r a n y x e T and w* e K*, choose x* ~ K - K* with d(x, x*) = d(x, K). By the definitions of K, K* a n d T, we have

cr(G) - 1 < d(x,x*) = d(x,w*) - 1 (2.1)

a n d

cr(G) = d(w, w*) = d(w, x*) - 1. (2.2)

C h o o s e shortest paths P(x,x*), P(w,w*), P(x,w) from x to x*, w to w*, and x to w respectively as in Fig. 2; where y (resp. z) is the vertex in P(x, x*)N P(x, w) (resp. P(w, w*) N P(x, w)) with largest distance f r o m x (resp. w).

X

y

2~**

Z

W

Fig. 2.

Suppose y = x*, then d(G) >_ d(x, w) = d(x,x*) + d(x*, w) >_ 2cr(G) by (2.1) a n d (2.2), a contradiction. Hence y ~ x*. Similarly, z v ~ w*. In the cycle (y . . . x*, w* . . . . ,z . . . y), by L e m m a 2.2, there is a vertex x** adjacent to b o t h x* a n d w*. N o t e that x** is n o t between y and x* (similarly, n o t between z a n d w*) otherwise d(x, w*) < d(x, x*), which contradicts (2.1). (2.1) also implies

Centers of Chordal Graphs 309 and (2.2) implies

d(w, x**) >_ d(w, x*) - 1 >_ cr(G). (2.4) (2.3) and (2.4) together with the assumption 2cr(G) - 1 >_ d(G) > d(x, w) imply that all inequalities in (2.3) and (2.4) are in fact equalities; and so x** e Bet(x*, 1, w).

N o w consider K** = K* U {x**: x e T}. It is easy to see that e(K**) <_ e(K). Since K** _~ Bet(K - K*, 1, w), by Theorem 2.6, K** is a clique. So K** is a clique center; in fact, is one with S(K**) ~ S(K) and d(w, w**) = cr(G) for all w** e K**. The same arguments as in the second p a r a g r a p h lead to a contradiction. This shows that d(G) >_ 2cr(G) and completes the proof of the theorem. [] As a consequence of Theorems 2.1 and 2.5, and the trivial inequalities cr(G) < r(G) < cr(G) + 1, for any connected chordal graph G exactly one of the follow- ing holds: 2r(G) = d(G) = 2cr(G), 2r(G) - 1 = d(G) = 2cr(G) + 1 and 2r(G) - 2 = d(G) = 2cr(G). F o r the case of block graphs, the last case is impossible since a block graph contains no 3-sun. F o r a block graph G with d(G) = 2cr(G) (resp. d(G) = 2cr(G) + 1), C(G) is a cut-vertex (resp. block); in any case C(G) is always a clique center.

3. Graphs Which Are Centers of Chordal Graphs

A chord of a path is an edge joining two nonconsecutive vertices of the path. L e m m a 3.1. In a chordal graph G, all vertices of a chordless path joining two vertices of C(G) entirely belongs to C(G).

Proof. Suppose x, y ~ C(G) and P(x, y) = (x = Vo, v l , . . . , v, = y) is a chordless x-y path. Let vi be the first vertex of P(x, y) which is not in C(G). Choose a vertex z such that d(vi, z) > r(G) and j > i as small as possible with d(vi, z) < r(G). Then d(Vk, z) > r(G) for i < k < j and d(vi_ 1, z) = d(vj, z) = r(G). Let w be the last c o m m o n vertex on shortest paths P(z, vi_l ) and P(z, vj). Then C = P(w, vi_~ ) U P(vi-~ , vj) U P(v~, w) is a cycle. By L e m m a 2.2, C has a vertex u adjacent to both vi-~ and v~. Since P(x, y) is chordless, u E P(w, vi-1) or u ~ P(vj, w). In the former case, d(z, vi) < d(z, u) + 1 = r(G) which is impossible. In the latter case, r(G) < d(z, vg) <_ d(z, u) + 1 < d(z, v~) -

1 + 1 = r(G), a contradiction. So the lemma holds. []

Theorem

3.2. The center of a connected chordal graph G is a distance invariance induced subgraph of G.

Proof. Since any shortest x-y path is chordless, the theorem follows from L e m m a

3.1. []

Theorem

3.3. The center of a connected chordal graph G is biconnected.

Proof. Suppose z is a cut vertex of the center (C(G)). Let x and y be two vertices in different components of ( C ( G ) ) - z . There are two disjoint x-y paths in G since C(G) lies in a biconnected component of G as shown in [7]. Take chords, if

there is any, to s h o r t e n these two p a t h s until two c h o r d l e s s

x-y

p a t h s are found. By L e m m a 3.1, all vertices of these two p a t h s are in

C(G).

T h e s e two p a t h s t h e n b o t h c o n t a i n the vertex z, a c o n t r a d i c t i o n . So the c e n t e r is b i c o n n e c t e d . [ ] N o t e t h a t T h e o r e m 3.3 was p r o v e d for 2-trees in [11]. By T h e o r e m 2.5, for a n y c o n n e c t e d c h o r d a l g r a p h G, there are three cases:

d(G)

= 2r(G),

d(G)

= 2r(G) - 1,

d(G)

= 2 r ( G ) - 2. W e shall d e r i v e s o m e r e s t r i c t i o n s o n d i a m e t e r s a n d r a d i i of centers of c h o r d a l g r a p h s a c c o r d i n g to these cases.

T h e o r e m 3.4.

C(G) is a clique for any connected chordal graph G with d(G)

= 2r(G).

Proof.

C h o o s e two vertices x a n d y such t h a t

d(x, y) = d(G).

F o r a n y

z ~ C(G),

we h a v e

2r(G) =

d(G) = d(x, y) < d(x, z) + d(z, y) < r(G) + r(G),

which i m p l y t h a t

d ( x , z ) = d ( z , y ) = r(G)

a n d so

z ~ Bet(x,r(G),y).

T h u s

C(G)

Bet(x, r(G), y).

T h e t h e o r e m then follows f r o m L e m m a 2.3. [ ]

I n general,

C(G)

is n o t necessarily a clique for the case of

d(G)

= 2r(G) - 1 o r 2r(G) - 2, see Fig. 3.

k

O

(a)

d(G)

= 2r(G) - 1 = 5 and

d(C(G)) = 3.

(b)

d(G)

= 2r(G) - 2 = 2 and

d(C(G)) = 2.

Fig. 3. Black vertices form

C(G)

Centers of Chordal Graphs 311

T h e o r e m 3.5.

d(C(G)) < 3 for any connected chordal 9raph G with d(G)

= 2r(G) - 1.

Proof.

C h o o s e x a n d y such t h a t

d(x, y) = d(G).

F o r a n y z ~ C(G), we h a v e

2r(G) - 1 =

d(G) = d(x, y) <_ d(x, z) + d(z, y) <

2r(G).

H e n c e

d(x, z)

a n d

d(z, y)

are e i t h e r

r(G)

- 1 o r

r(G)

b u t n o t b o t h

r(G)

- 1. W e shall p r o v e t h a t e i t h e r

z ~ Bet(x, r(G), y)

o r is a d j a c e n t to s o m e vertex in

Bet(x, r(G), y).

Since

Bet(x,r(G),y)

is a clique b y L e m m a 2.3, every two vertices o f

C(G)

are of d i s t a n c e at m o s t three in G a n d hence in

C(G).

F o r the case o f

d(x, z) = r(G)

a n d

d(z, y) = r(G) - 1, z ~ Bet(x, r(G), y).

F o r the case o l d ( x , z) =

r(G)

- 1 a n d

d(z, y) = r(G), z ~ Bet(x, r(G) - 1, y)

a n d so is a d j a c e n t to s o m e vertex in

Bet(x,

r(G), y).

S u p p o s e

d(x, z) = d(z, y) = r(G).

C h o o s e s h o r t e s t p a t h s

P(x, y), P(y, z), P(z, x)

which p a i r w i s e m e e t at x*, y*, z* as in Fig. 4. Since

r(G) > d(x,

t), r(G) >_

d(z, y)

a n d

d(x,z*) + d(z*,y) > d(x,y) >_

2r(G) - 1, we m u s t have z = z*. By L e m m a 2.2, there is a vertex w in the cycle C = ( x * , . . .

,u,z*,v ... y* ... x*)

which is a d j a c e n t to b o t h z* a n d v. If w =

u,

t h e n

d(x, u) = d(v,y)

= r(G) - 1 a n d (u, v) ~ E i m p l y

v ~ Bet(x, r(G),

y); a n d so z is a d j a c e n t to vertex v in

Bet(x, r(G), y).

If w ~ u, then w is b e t w e e n x* a n d y* as in Fig. 4. N o t e t h a t

r(G) = d(x,z*) < d(x,w)

+ 1, i.e.

d(x, w) >_ r(G)

- 1. S i m i l a r l y

d(w, y) >_ r(G)

- 1. So

d(x, w) = r(G)

o r

r(G)

- 1. In the f o r m e r case, z is a d j a c e n t to

w E Bet(x, r(G), y).

In the l a t t e r case, z is a d j a c e n t to

v ~ Bet(x, r(G), y).

This c o m p l e t e s the p r o o f o f the t h e o r e m . [ ] S i m i l a r a r g u m e n t s as in the p r o o f o f T h e o r e m 3.5 l e a d to the following result.

z, ~ ~ z

27 a ; * 'tO y * f f

Fig. 4.

T h e o r e m 3.6.

d(C(G)) < 5 for any connected chordal 9raph G with d(G)

= 2r(G) - 2. By o b s e r v i n g m a n y e x a m p l e s , we h a v e the following conjecture.

Conjecture: d(C(G))

< 2 for a n y c o n n e c t e d c h o r d a l g r a p h G with

d(G)

= 2r(G) - 2. N e x t we s t u d y sufficient c o n d i t i o n s for a b i c o n n e c t e d c h o r d a l g r a p h H to be the c e n t e r o f s o m e c h o r d a l g r a p h G. I f d ( H ) = 1 o r

d(H) = r(H)

= 2, then H is the c e n t e r o f itself. F o r the case of

d(H)

= 2 a n d

r(H)

= 1, we h a v e the f o l l o w i n g result b y using the m a i n t h e o r e m o f S e c t i o n 2.

Theorem 3.7. Suppose H = ( U , F ) is a biconnected chordal 9raph with d ( H ) = 2, r(H) = 1 and x E C(H). H is the center of some chordal 9raph G = (V, E) if and only if d ( H - x) <_ 3.

Proof. ( ~ ) S u p p o s e H is the c e n t e r of G, i.e. U = C(G). C h o o s e w e V such t h a t d(x, w) = r(G). F o r a n y z e U - x, z is a d j a c e n t to x a n d d(z, w) < r(G). H e n c e either d(z, w) -- r(G) - 1 a n d so z e Bet(x, 1, w), o r else d(z, w) = r(G). F o r the l a t t e r case, d(z, w) = d(x, w) = r(G) i m p l y , b y L e m m a 2.4, t h a t Bet(z, 1, w) N Bet(x, 1, w) ~ ;0 a n d so z is a d j a c e n t to s o m e vertex in Bet(x, 1, w). T h i s is true for all z e H. Since Bet(x, 1, w) is a cfique, d(H - x) < 3.

( ~ ) S u p p o s e d(H - x) < 3. By T h e o r e m 2.1, cr(H - x) = 1. Let K be a clique c e n t e r of H - x. T h e n every vertex in U - x is a d j a c e n t to x, a n d every vertex in U - K is a d j a c e n t to s o m e vertex in K. C o n s i d e r the g r a p h G o b t a i n e d f r o m H b y a d d i n g two new vertices u a n d v such t h a t u is a d j a c e n t to x a n d v is a d j a c e n t to all vertices of K. It is s t r a i g h t f o r w a r d to check t h a t G is a c h o r d a l g r a p h a n d H is the

c e n t e r of G. [ ]

W e close this p a p e r b y the following s u m m a r y of results: a g r a p h H is the c e n t e r of s o m e c h o r d a l g r a p h if a n d o n l y if (1) H is c h o r d a l a n d b i c o n n e c t e d a n d (2) d ( n ) = 1, o r d ( n ) = r(H) = 2, o r d(H) = 2, r(H) = 1 a n d d ( n - x) < 3 for a n y x ~ C(H), o r d(H) = 3, r(H) = 2 a n d " s o m e c o n d i t i o n s we still d o n o t k n o w " , o r d(H) = 4 o r 5 (we c o n j e c t u r e t h a t this case is impossible).

Acknowledgement, The author wishes to express his gratitude to refree for many useful suggestions about the revision of the paper.

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Received: March 30, 1989 Revised: November 22, 1990