Centers and medians ofdistance-hereditary
graphs
Hong-Gwa Yeh
a, Gerard J. Chang
b;∗;1aDepartment of Mathematics, National Central University, Chung-Li 320, Taiwan bDepartment of Mathematics, National Taiwan University, Taipei 106, Taiwan Received 22 August 2000; received in revised form 8 May 2002; accepted 28 May 2002
Abstract
A graph is distance-hereditary ifthe distance between any two vertices in a connected induced subgraph is the same as in the original graph. In this paper, we study metric properties of distance-hereditary graphs. In particular, we determine the structures ofcenters and medians of distance-hereditary and related graphs. The relations between eccentricity, radius, and diameter ofsuch graphs are also investigated.
c
2002 Elsevier Science B.V. All rights reserved.
Keywords: Distance; Eccentricity; Diameter; Radius; Center; Median; Distance-hereditary graph; Chordal graph; Ptolemaic graph
1. Introduction
This paper investigates metric properties ofcenters and medians ofdistance-hereditary graphs.
Suppose G = (V; E) is a graph with vertex set V and edge set E. The distance
dG(x; y) or d(x; y) between two vertices x and y in the graph G is the minimum
number ofedges ofan x–y path in G. The eccentricity eG(v) ofa vertex v in G is
maxx∈Vd(v; x). The diameter diam(G) of G is the largest eccentricity ofa vertex in
G, and the radius rad(G) is the smallest. The center of G is the set C(G) = {v∈V : eG(v)6eG(x) for all x∈V }:
∗Corresponding author.
E-mail addresses:hgyeh@math.ncu.edu.tw(H.-G. Yeh),gjchang@math.ntu.edu.tw(G.J. Chang). 1Supported in part by the National Science Council under grant NSC87-2125-M009-007. 0012-365X/03/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S0012-365X(02)00630-1
The distance sum DG(v) of v is x∈VdG(v; x). The median of G is the set
M(G) = {v∈V : DG(v)6DG(x) for all x∈V }:
Suppose w is a real-valued function on V . The w-distance sum DG; w(v) of v is
x∈VdG(v; x)w(x). The w-median of G is the set
Mw(G) = {v∈V : DG; w(v)6DG; w(x) for all x∈V }:
The local w-median of G is the set
LMw(G) = {v∈V : DG; w(v)6DG; w(x) for all x adjacent to v}:
We very often also call the subgraph induced by the center (respectively, median, w-median, local w-median) simply the center (respectively, median, w-median, local w-median) ofthe graph.
The problem ofdetermining shapes ofcenters and medians for di@erent classes of graphs have been extensively studied in the literature, see Refs. [1,3–5,6,10,11,14,15,18–
42]. The earliest such kind ofresult due to Jordan [18] is that the center or the median ofa tree is either a single vertex or two adjacent vertices. Slater [30,31] examined the structure ofvariations ofcenters for trees. Proskurowski [26] proved that the center of a maximal outplanar graph is one ofseven special graphs. As a generalization, he [27] found all possible centers of2-trees and showed that the center ofa 2-tree is bicon-nected. Laskar and Shier [19] proved that the center ofa connected chordal graph is connected. Chang [5] showed that the center ofa connected chordal graph is distance invariant and biconnected. He also gave a characterization ofa biconnected chordal graph ofdiameter 2 and radius 1 to be the center ofsome chordal graph. Yushmanov [38,40] showed that the center ofa connected chordal graph is m-convex and is either a complete graph or its connectivity is no smaller than the connectivity ofthe block in which it lies. Soltan and Chepoi [36] proved that the center ofa connected chordal graph has diameter at most 3.
Slater [31] studied the structure ofvarious types ofmedians for trees. He [32] also proved that for every graph H there exists a graph G whose median is H, and that the median ofa 2-tree is isomorphic to K1, K2 or K3. Yushmanov [41] and Nieminen [25]
showed that the median ofa Ptolemaic graph is a complete graph. Lee and Chang [20] proved that the w-median ofa connected strongly chordal graph is a complete graph when the function w is positive. Wittenberg [37] proved that for any chordal graph the local w-median coincides with the w-median ifa certain neighborhood condition holds.
Many results on centers and medians are on graphs with tree-like structures. In this paper, we focus on distance-hereditary graphs, which include trees and cographs. The
paper is organized as follows. In Section 2, we survey some properties
ofdistance-hereditary graphs. We also introduce the concept ofconcavity ofa path, and derive two lemmas that are useful in this paper. Section3 shows that the center ofa distance-hereditary graph (respectively, bipartite distance-distance-hereditary graph) is either a connected graph with diameter at most 3 or a cograph (respectively, an independent set of G).
function w is a cograph, and the w-median ofa bipartite distance-hereditary graph G with a positive function w is either a connected cograph or an independent set of G. In Section 5, we show that the w-median ofa distance-hereditary graph with a positive function w “nearly” coincides with its local w-median.
2. Preliminaries of distance-hereditary graphs
This section gives a briefintroduction to distance hereditary and related graphs. We also introduce the concept ofconcavity ofa path, and give two related lemmas that are useful in the paper.
Suppose S is a vertex subset ofa graph G = (V; E). Denote G[S] the subgraph of G induced by S. The deletion of S from G, denoted by G − S, is the induced subgraph
G[V −S]. The neighborhood NG(v) or N(v) ofa vertex v is the set ofall vertices
adjacent to v. An induced (or chordless) path is a path v1; v2; : : : ; vn in which vi is not
adjacent to vj whenever |i − j| = 1. A vertex subset S ofa graph G is called m-convex
ifall the vertices ofall the induced path joining vertices ofS lie in S. Two vertices x and y are connected ifthere is an x–y path in G; otherwise they are disconnected.
The distance between two vertex subsets A and B is dG(A; B) = min{dG(a; b): a∈A
and b∈B}.
The hanging hu ofa connected graph G = (V; E) at a vertex u∈V is the
collec-tion ofsets L0(u); L1(u); : : : ; Lt(u) (or L0; L1; : : : ; Lt ifthere is no ambiguity), where
t = maxv∈VdG(u; v) and Li(u) = {v∈V : dG(u; v) = i} for 06i6t. Li(u) is called the
level i ofthe hanging hu. For any 16i6t and any vertex v∈Li, let N(v) = N(v) ∩ Li−1.
A graph G is distance hereditary ifeach connected induced subgraph F of G has the property that dF(u; v) = dG(u; v) for every pair of vertices u and v in F.
Distance-hereditary graphs were introduced by Howorka [16]. The characterizations and recogni-tions ofdistance-hereditary graphs have been studied in [2,8,13,16]. A graph is chordal ifevery cycle oflength greater than three has a chord. A cograph is a graph containing no induced path offour vertices, see [7]. A graph is Ptolemaic if for any four vertices x, y, z, w ofit, the Ptolemy inequality d(x; y)d(z ; w)6d(x; z)d(y; w) + d(x; w)d(y; z) holds. It was shown in [17] that G is Ptolemaic ifand only ifG is distance hereditary and chordal. Some containment relationships among these and other families of graphs are as follows:
trees ⊂ block graphs ⊂ Ptolemaic graphs ⊂ distance-hereditary graphs, trees ⊂ bipartite distance-hereditary graphs ⊂ distance-hereditary graphs, cographs ⊂ distance-hereditary graphs,
Ptolemaic graphs ⊂ strongly chordal graphs ⊂ chordal graphs.
Theorem 1 (Bandelt and Mulder [2], D’Atri and Moscarini [8], Hammer and Ma@ray
[13], Howorka [16]). For any connected graph G = (V; E) the following statements are equivalent:
(1) G is a distance-hereditary graph.
(3) Every induced path in G is a shortest path.
(4) For every hanging hu= (L0; L1; : : : ; Lt) of G and every pair of vertices x, y∈Li
(16i6t) that are in the same component of G − Li−1, N(x) = N(y).
Theorem 2 (Bandelt and Mulder [2], D’Atri and Moscarini [8], Hammer and Ma@ray
[13], Howorka [16]). Suppose hu= (L0; L1; : : : ; Lt) is the hanging of a connected
dis-tance-hereditary graph G at u. Then, every Li induces a cograph. Moreover, if G is
bipartite, then every Li is an independent set of G.
Suppose G = (V; E) is a graph. For two vertices x and y in V , let Vxy= {v∈V : dG(x; v)¡dG(y; v)}:
DeMne the function ‘u from V to non-negative integers by: ‘u(v) = k whenever
dG(u; v) = k, or equivalently, v is in Lk for the hanging hu= (L0; L1; : : : ; Lt) of G at u.
A path P : x0; x1; : : : ; xk is u-concave downward if ‘u(x0)¿‘u(x1)¿ · · · ¿‘u(xr−1)¿
‘u(xr) = ‘u(xr+1) = · · · = ‘u(xr)¡‘u(xr+1)¡ · · · ¡‘u(xk−1)¡‘u(xk) for some 06r6 r6k with 06r− r61. P is monotone if r = k or r= 0.
The following two lemmas are useful in this paper. Note that they are trivial for the case when the distance-hereditary graphs are trees.
Lemma 3. Suppose hu= (L0; L1; : : : ; Lt) is the hanging of a connected
distance-hereditary graph G at u. For any two vertices x and y in G, there exists a shortest x–y path which is u-concave downward.
Proof. Suppose P : x = x0; x1; : : : ; xk= y is a shortest x–y path, where dG(x; y) = k. We
may assume that P is chosen such that s(P) = kj=0‘u(xj) is as small as possible. Let
i be the largest index such that x0; x1; : : : ; xi is u-concave downward. Note that i¿1.
In fact i = k and so the lemma holds. Suppose to the contrary that i¡k. Then one of the following cases holds:
(1) ‘u(xi−1) = ‘u(xi)¿‘u(xi+1).
(2) ‘u(xi−1)¡‘u(xi) = ‘u(xi+1).
(3) ‘u(xi−1) = ‘u(xi) = ‘u(xi+1).
(4) ‘u(xi−1)¡‘u(xi)¿‘u(xi+1).
For case (1) or (2), by Theorem 1 (4), xi−1 is adjacent to xi+1, a contradiction to that
P is a shortest path. For case (3) or (4), by Theorem 1 (4), xi−1 and xi+1 have a
common neighbor x
i with ‘u(xi) = ‘u(xi−1) − 1. Then, the path P resulting from P by
replacing xi with xi is also a shortest x–y path whose s(P)¡s(P), a contradiction to
the choice of P. This completes the proofofthe lemma.
Lemma 4. Suppose P : x0; x1; : : : ; xk is an induced path of a distance-hereditary graph
G = (V; E). If k = 2 with G chordal or k¿3, then Vx0x1 is a proper subset of Vxk−1xk. Proof. Suppose hx0= (L0; L1; : : : ; Lt) is the hanging of G at x0. Assume that there exists some vertex v∈Vx0x1−Vxk−1xk. By Lemma3, there exists an x0-concave downward
shortest v–xkpath P: v = y0; y1; : : : ; yr; : : : ; yr; : : : ; yk= xk, where yr (respectively, yr) is the Mrst (respectively, last) vertex of P in a smallest level Lf with 06r− r61.
For the case when f¿2, yr and xf are in the same component of G−Lf−1.
Accord-ing to Theorem 1 (4), yr is adjacent to xf−1 and so v = y0; y1; : : : ; yr; xf−1; xf−2; : : : ; x1;
x0 is a shortest v–x0 path, contradicting that v∈Vx0x1.
For the case when f = 0, v = y0; y1; : : : ; yr; x1; x2; : : : ; xk−1; xk is a shortest v–xk path,
contradicting that v∈Vxk−1xk.
Now, suppose f = 1. For the case when k¿3, yr+1 and x2 are in the same compo-nent of G −L1. According to Theorem 1 (4), yr is adjacent to x2 and so v = y0; y1; : : : ; yr; x2; x3; : : : ; xk−1; xk is a shortest v–xk, contradicting that v∈Vxk−1xk. For the case when k = 2 and G is chordal, yr and x1 adjacent to x2 imply that they are also adjacent to x0, i.e., yrx0x1x2yr is a cycle of G. By the deMnition ofa chordal graph, yr is adja-cent to x1. If r = r, then dG(v; x1)6dG(v; x0), contradicting that v∈Vx0x1. If r = r− 1, then x0x1x2yryrx0 is a cycle of G. By Theorem 1 (2), yr is adjacent to x1, and hence dG(v; x1)¡dG(v; x2) contradicting that v∈Vxk−1xk.
Therefore, Vx0x1 is a subset of Vxk−1xk; and in fact a proper subset as
xk−1∈Vxk−1xk− Vx0x1. This completes the proofofthe lemma. 3. Centers
The purpose ofthis section is to investigate the shapes ofcenters ofdistance-heredit-ary graphs. We in fact study centers in a more general setting as follows. Suppose S is a non-empty subset of V in a graph G = (V; E). The S-eccentricity eG; S(v) ofa vertex
v in G is maxx∈S d(v; x).
The S-center of G is CS(G) = {v∈V : eG; S(v)6eG; S(x) for all x∈V }.
The anticenter of G is AC(G) = {v∈V : eG(v)¿eG(x) for all x∈V }.
The S-anticenter of G is ACS(G) = {v∈V : eG; S(v)¿eG; S(x) for all x∈V }.
Theorem 5. Suppose S is a non-empty vertex set of a distance-hereditary graph G = (V; E). If H is a connected component of G[T], where T ⊆ V with eG; S(x) = eG; S
(y) for every two vertices x and y in T, then diam(H)63. If moreover G is Ptolemaic, then diam(H)62.
Proof. Suppose x and y are two vertices in V (H) such that dH(x; y) = diam(H) = k.
Choose an induced x–y path P : x = x0; x1; : : : ; xk= y in H. Note that P is also an
induced path of G and eG; S(x0) = eG; S(x1) = · · · = eG; S(xk). Let eG; S(x1) = dG(x1; z) f or
some vertex z ∈S. Then, for 06i6k, we have
dG(xi; z)6eG; S(xi) = eG; S(x1) = dG(x1; z); i:e:; z ∈Vxix1 or dG(xi; z) = dG(x1; z):
We Mrst prove that k63. Suppose to the contrary that k¿4. If z ∈Vx0x1, then
z ∈Vx2x3and z ∈Vx3x4 by Lemma4. These imply that eG; S(x4)¿dG(x4; z) = dG(x3; z)+1 = dG(x2; z)+2¿dG(x1; z) = eG; S(x1), a contradiction. Hence dG(x0; z) = dG(x1; z). We then
hang G at z. Note that x0 and x1 are in the same level. If x2 is in level dG(x2; z) =
dG(x1; z)6dG(x2; z). Since dG(x0; z) = dG(x1; z), we have z ∈Vx1x0. By Lemma4, z ∈Vx3x2 and so dG(x2; z)6dG(x3; z). Therefore, dG(x0; z) = dG(x1; z)6dG(x2; z)6dG(x3; z)
6dG(x1; z) and so dG(x0; z) = dG(x1; z) = dG(x2; z) = dG(x3; z). Now, consider the
hang-ing ofgraph G at z. Then x0; x1; x2; x3 are in the same level. By Theorem 1 (4), there
exists a vertex z∗ adjacent to x0, x1, x2, and x3. Note that z∗x0x1x2x3z∗ is a cycle of length 5 without crossing chords, a contradiction. Hence k63, i.e., diam(H)63.
Next, we prove that k62 when G is Ptolemaic. Suppose to the contrary that k¿3. If z ∈Vx0x1, then z ∈Vx1x2 by Lemma 4. This implies that eG; S(x2)¿dG(x2; z)¿dG(x1; z) = eG; S(x1), a contradiction. Hence z ∈Vx0x1 and so dG(x0; z) = dG(x1; z) and z ∈Vx1x0. Then, by Lemma4, z ∈Vx2x1and z ∈Vx3x2. Therefore, dG(x0; z) = dG(x1; z)6dG(x2; z)6dG(x3; z) 6dG(x1; z) and so dG(x0; z) = dG(x1; z) = dG(x2; z) = dG(x3; z). Again, we can get a cycle
z∗x
0x1x2x3z∗ oflength 5 without crossing chords, a contradiction. Hence k62, i.e.,
diam(H)62.
Corollary 6. Suppose S is a non-empty vertex set in a distance-hereditary graph G.
If H is a connected component of the subgraph induced by CS(G) or ACS(G), then
diam(H)63. If moreover G is Ptolemaic, then diam(H)62.
The distance-hereditary graphs G1 and G2 and the Ptolemaic graphs G3
and G4 in Fig. 1 show that the bounds in Corollary 6 are sharp. Note that C(G1) =
{a1; b1; c1; d1; e1; f1}, AC(G2) has a connected component G2[{a2; b2; c2; d2}], C(G3) =
{a3; b3; c3; d3; e3}, and AC(G4) has a connected component G4[{a4; b4; c4; d4; e4}].
Because distance-hereditary graphs have a “tree like” structure ofadjacency, one may expect that their centers are “small” and “compact”. The following lemma supports such expectations.
Lemma 7. If V1 and V2 are the vertex sets of two distinct components of the S-center
of a distance-hereditary graph G = (V; E), then dG(V1; V2) = 2.
Proof. Assume that dG(V1; V2)¿3. Then there exists an induced path P : x0; x1; : : : ; xk,
where k¿3, x0∈V1, xk∈V2, but x1; xk−1∈CS(G). Assume z is a vertex in S with
dG(xk−1; z) = eG; S(xk−1). Since xk∈CS(G) and xk−1∈CS(G), dG(xk; z)6eG; S(xk)¡eG; S
(xk−1) = dG(xk−1; z) and so z ∈Vxkxk−1. By Lemma 4, we then have z ∈Vx1x0. Let hz= (L0; L1; : : : ; Lt) be the hanging of G at z. Since x0∈CS(G) and xk−1∈CS(G), we have
dG(x0; z)6eG; S(x0)¡eG; S(xk−1) = dG(xk−1; z). Therefore, the relative positions of x0; x1;
xk−1; xk are as shown in Fig. 2(a). Thus, there exists a vertex xj in the path P such
that xj and xk are in the same level, say Li, of hz, and xjxj+1xj+2: : : xk is a path in
G − Li−1 (see Fig. 2(b)). Then, by Theorem 1 (4), xk is adjacent to xj−1, contrary to
that P is an induced path. Therefore, dG(V1; V2)62.
Theorem 8. If G is a distance-hereditary graph, then the S-center CS(G) is either a
connected graph of diameter 3 or a cograph. If moreover G is a bipartite
distance-hereditary graph, then the S-center CS(G) is either a connected graph of diameter
Fig. 1. Examples for which the bounds in Corollary6are sharp.
Proof. First, if H = G[CS(G)] is connected, then the theorem follows immediately
from Theorem 5. Hence, we may assume that H is disconnected. Choose any two
distinct components H1 and H2. Then, by Lemma 7, there exists an induced path
xzy in G such that x∈V (H1), y∈V (H2) and z ∈CS(G). Suppose w is a vertex in S
with dG(w; z) = eG; S(z). Then dG(w; x)6eG; S(x)¡eG; S(z) = dG(w; z) and so dG(w; z) =
dG(w; x) + 1. Similarly, dG(w; z) = dG(w; y) + 1.
Let hw= (L0; L1; : : : ; Lt) be the hanging of G at w. Note that x and y lie on the same
level Li of hw and z ∈Li+1. Now for every vertex x∈H1, dG(w; x)6eG; S(x)¡eG; S(z)
= dG(w; z) and so x∈Lr for some r6i. In f act r = i. Suppose to the contrary that
x∈L
r for some r6i − 1. Then H1 has an x–x path P laying above level i + 1. Hence
P contains an edge uv such that y and u are in the same component of G − Li−1 and
v∈Li−1. Thus, by Theorem 1 (4), y is adjacent to v and hence y∈H1, a contradiction.
Therefore, every vertex of H1 lies on level Li. Hence H1 is a cograph by Theorem 2.
Moreover, if G is also bipartite, then Li is an independent set of G by Theorem 2, and
so is H1. This completes the proofofthe theorem.
As described in [5], Hedetniemi proved that any graph H is isomorphic to the center ofsome graph G ofdiameter 4 and radius 2. When H is a cograph, an analogous
Fig. 2. For the proofofLemma 7.
result for distance-hereditary graph G is the following theorem. Using the proofofthis theorem, we can construct a distance-hereditary graph whose center induces a graph with arbitrary number ofcomponents. Also the center ofa bipartite distance-hereditary graph can be an independent set ofarbitrarily large size.
Theorem 9. For any given cograph H there exists a connected distance-hereditary graph G whose center is isomorphic to H.
Proof. We construct G by adding four new vertices u, v, w, x into H such that v and w are adjacent to all vertices of H, u is adjacent only to v and x only to w. It is clear that G is a distance-hereditary graph whose center is isomorphic to H.
4. Medians
This section discusses the structures ofmedians ofdistance-hereditary graphs. We again study medians in a more general setting. Suppose S is a non-empty subset of V . The S-distance sum DG; S(v) is equal tox∈S dG(v; x). The S-w-distance sum DG; S; w(v)
of v is x∈S dG(v; x)w(x). The S-median (also called the S-centroid [31]) of G is the
set
The S-w-median of G is the set
MS; w(G) = {v∈V : DG; S; w(v)6DG; S; w(x) for all x∈V }:
The antimedian of G is the set
AM(G) = {v∈V : DG(v)¿DG(x) for all x∈V }:
Lemma 10 (Entringer et al. [10], Slater [31]). If a and b are two adjacent vertices of a graph G = (V; E) with a function w on V, then DG; w(a)−DG; w(b) = w(Vba)−w(Vab).
Proof. The lemma follows immediately from the fact that DG; w(a) − DG; w(b) = x∈Vab {dG(x; a) − dG(x; b)}w(x) − x∈Vba {dG(x; a) − dG(x; b)}w(x):
Lemma 11. Suppose P : x0; x1; : : : ; xk is an induced path of a distance-hereditary graph
G = (V; E) with a function w¿0 (respectively, w¿0) on V. If either k¿2 with G chordal or k¿3, then DG; w(x0)−DG; w(x1)¿ (respectively, ¿) DG; w(xk−1)−DG; w(xk).
Proof. The lemma follows immediately from Lemmas 4 and 10.
Theorem 12. For any S ⊆ V of a Ptolemaic graph G = (V; E) with a function w¿0, the S-w-median MS; w(G) is m-convex.
Proof. Assume MS; w(G) is not m-convex. Then there exists an induced path P : x0; x1;
: : : ; xk in G with k¿2 such that x0; xk∈MS; w(G) but x1; xk−1∈MS; w(G). Then, by
Lemmas 10 and 4, we have
0 ¿ DG; S; w(x0) − DG; S; w(x1)
= w(Vx1x0∩ S) − w(Vx0x1∩ S) ¿ w(Vxkxk−1∩ S) − w(Vxk−1xk∩ S)
= DG; S; w(xk−1) − DG; S; w(xk)¿0;
a contradiction. So, MS; w(G) is m-convex.
Corollary 13 (Soltan [35]). The median of a Ptolemaic graph is connected.
Corollary 14 (Slater [31]). For any subset S of the vertices of a tree T, the S-median of T is connected.
Theorem 15. Suppose G = (V; E) is a distance-hereditary graph with a function w¿0. If H is a connected component of G[T], where T ⊆ V having DG; w(x) = DG; w(y) for
every two vertices x and y in T, then H is a cograph. If moreover G is Ptolemaic, then H is a clique.
Proof. Suppose P : x0; x1; : : : ; xk is an induced path in H. Note that P is also an
induced path of G. If G is distance hereditary (respectively, Ptolemaic) and k¿3 (respectively, k¿2), then by Lemma 11 and the fact that w¿0, we have DG; w(x0) −
DG; w(x1)¿DG; w(xk−1) − DG; w(xk), contrary to DG; w(x0) = DG; w(x1) = DG; w(xk−1) =
DG; w(xk). So, k62 (respectively, 61) and hence H is a cograph (respectively, a
clique).
Corollary 16 (Nieminen [25], Yushmanov [38]). The median of a Ptolemaic graph is a clique.
Corollary 17 (Yushmanov [38]). Every connected component of the subgraph induced
by the antimedian of a Ptolemaic graph is a clique.
It is worth pointing out that the S-median ofa Ptolemaic graph does not have a theorem like Theorem15. As shown in [31], there exist trees whose S-median contains a path of n vertices for any n. The following theorem shows that the median of a distance-hereditary graph nearly coincides with its local median.
Theorem 18. Suppose G is a distance-hereditary graph with a function w¿0. If
x∈Mw(G) and y∈LMw(G), then d(x; y)62.
Proof. Suppose P : x = x0; x1; : : : ; xk= y is an induced x–y path of G. Assume k¿3. By
Lemma 11, we have 0¿DG; w(x0) − DG; w(x1)¿DG; w(xk−1) − DG; w(xk)¿0, a
contradiction. So dG(x; y)62.
Theorem 19. If G is a Ptolemaic graph with a function w¿0, then Mw(G) = LMw(G).
Proof. Assume that LMw(G) − Mw(G) = ∅. Pick y∈LMw(G) − Mw(G) and x∈Mw(G)
such that dG(x; y) = dG(LMw(G) − Mw(G); Mw(G)). Suppose P : x = x0; x1; : : : ; xk= y is
an induced x–y path of G. Note that k¿2 and x1∈Mw(G). Hence, by Lemma 11,
0¿DG; w(x0) − DG; w(x1)¿DG; w(xk−1) − DG; w(xk)¿0, a contradiction. Therefore, LMw
(G) − Mw(G) = ∅ and so Mw(G) − LMw(G).
Theorem 20. If G = (V; E) is a distance-hereditary graph with a function w¿0, then its w-median is a cograph. If moreover G is bipartite distance hereditary, then its w-median is either a connected cograph or an independent set of G.
Proof. The Mrst part of the theorem follows from Theorem15 immediately. To prove
the second part, suppose V1 and V2 are the vertex sets oftwo distinct components of
the w-median of G. For any two vertices x∈V1 and y∈V2, by Theorem 18, we have
dG(x; y) = 2. Now consider the hanging hx= (L0; L1; : : : ; Lt) of G at x. Clearly V2⊆ L2
5. Convexity and diameters
This section investigates metric properties for chordal graphs and distance-hereditary graphs.
A vertex subset S is called an x–y separator of G if x and y are in di@erent components of G − S. An x–y separator S is said to be minimal ifno proper subset of S is an x–y separator of G.
Theorem 21 (Dirac [9]). Every minimal x–y separator of a chordal graph is a clique. Theorem 22. If G = (V; E) is a chordal graph and S ⊆ V , then the S-center CS(G) of
G is m-convex.
Proof. Assume to the contrary that CS(G) is not m-convex. Then there exist x and y in
CS(G) with an induced x–y path P of G such that |V (P)|¿3 and P ∩ CS(G) = {x; y}.
Suppose C is a minimal x–y separator of G. So, there exists a vertex r ∈P ∩ C and hence r ∈C − CS(G). Suppose s∈S with dG(s; r) = eG; S(r). If s∈C then eG; S(r) = 1
contradicting that r ∈CS(G). Hence, without loss ofgenerality, we may assume
that s and x are in di@erent components of G − C. By the fact that C is a
clique, there exists a vertex t ∈C such that eG; S(x)¿dG(s; x) = dG(s; t) + dG(t; x)¿
dG(s; t) + 1¿dG(s; r) = eG; S(r) contrary to that eG; S(x)¡eG; S(r). Therefore, CS(G) is
m-convex.
Corollary 23 (Yushmanov [38,40]). The center C(G) of a chordal graph is m-convex. Suppose x and y are two vertices in a graph G with dG(x; y) = eG(y). It is easily seen
that if G is a tree then eG(x) = diam(G). In general, eG(x) = diam(G) for a
distance-hereditary graph G. Moreover, the di@erence between eG(x) and diam(G) may be
arbitrarily large for a general graph. However, in the following theorem we show that eG(x) is nearly equal to diam(G) for a distance-hereditary graph G. The graphs given
in Fig. 3 show that the bounds in the following theorem are sharp.
Theorem 24. For any vertex y in a distance-hereditary graph G = (V; E). If x is a vertex with dG(y; x) = eG(y), then eG(x)¿diam(G) − 2. If moreover G is Ptolemaic,
then eG(x)¿diam(G) − 1.
Proof. Let hu= (L0; L1; : : : ; Lt) be the hanging of G at a vertex u having eG(u) = diam
(G) = t. Choose a vertex z in Lt. By Lemma 3, there exists a u-concave downward
shortest x–y path
Px: x = x0; x1; : : : ; xr; : : : ; xr; : : : ; xk= y;
where xr (respectively, xr) is the Mrst (respectively, last) vertex of Px in a smallest
level Lfx with 06r− r61; and a u-concave downward shortest z–y path
Pz: z = z0; z1; : : : ; zs; : : : ; zs; : : : ; zm= y;
where zs (respectively, zs) is the Mrst (respectively, last) vertex of Pz in a smallest level Lfz with 06s− s61.
We may assume that ‘u(z)−2¿‘u(x), for otherwise diam(G)−1 = ‘u(z)−16‘u(x)6
eG(x) and so the theorem holds.
Suppose xi= zj for some i6r and j6s. Since dG(zj; z) = ‘u(z) − ‘u(zj)¿‘u(x) −
‘u(xi) = dG(xi; x), we have dG(y; z) = dG(y; zj) + dG(zj; z)¿dG(y; xi) + dG(xi; x) =
dG(y; x) = eG(y), a contradiction. Therefore, the two paths x0; x1; : : : ; xr and z0; z1; : : : ; zs
have no vertex in common.
Next, (‘u(x) − fx) + (r− r) + (‘u(y) − fx) = dG(x; y) = eG(y)¿dG(z ; y) = (‘u(z) −
fz) + (s− s) + (‘u(y) − fz). Therefore, ‘u(z) − 2¿‘u(x) and r− r61 and 06s− s
imply fx¡fz. Let xq (respectively, xq) be the Mrst (respectively, last) vertex of Px in level Lfz−1. Since xq+1 and zs are connected in G − Lfz−1, by Theorem 1 (4), xq is adjacent to zs. Consider the x–z path
P1: x = x0; x1; x2; : : : ; xq; zs; zs−1; : : : ; z0= z:
Suppose P1 is an induced path. Note that dG(x; xq) + dG(xq; y) = dG(x; y) = eG(y)¿dG(u; y) = dG(u; xq) + dG(xq; y). Then, dG(x; xq)¿dG(u; xq) and so eG(x)¿ dG(x; z) = dG(x; xq) + dG(xq; z)¿dG(u; xq) + dG(xq; z)¿dG(u; z) = diam(G). In this case, the theorem holds.
We then may assume that P1 is not an induced path, say P1 has a chord joining
some vertex xi to some vertex zj. Note that in this case 0¡q6r6r6q¡k. Then,
either i = q with j = s, or i6q−1 with j6s. For the Mrst case, xqzs∈E. For the second
case, xq−1 and zs are connected in G − Lfz−1, and so again xqzs∈E by Theorem 1 (4). In any case, dG(zs; xq) = 1.
Since dG(y; zs) + dG(zs; xq) + dG(xq; x) ¿ dG(y; x) = eG(y) ¿ dG(y; z) =
dG(y; zs) + dG(zs; z), we have 1 + dG(xq; x)¿dG(zs; z). By the fact that dG(xq; x) =
‘u(x) − ‘u(xq)6eG(x) − (fz − 1) and dG(zs; z) = diam(G) − fz, we then have
eG(x)¿diam(G) − 2. This proves the Mrst part ofthe theorem.
To prove the second part ofthe theorem, suppose G is Ptolemaic, i.e., G is chordal and distance hereditary. For the case when xq= xq, we have q = r = r= q and fx= fz − 1. For the case when xq= xq, since the two vertices xq and xq in Lfz−1 are adjacent to zs∈Lfz, they are also adjacent to some w∈Lfz−2 according to Theorem 1 (4). By the chordality of G, the cycle w; xq; zs; xq; w has a chord, which must be xqxq. So, q = r¡r= q and fx= fz− 1. In any case, dG(xq; xq)61. Consider the x–z path
Suppose P2 is an induced path. Note that dG(x; xq) + dG(xq; xq) + dG(xq; y) = dG(x; y) = eG(y)¿dG(u; y) = dG(u; xq) + dG(xq; y) and so dG(x; xq)¿dG(u; xq) − 1 since dG(xq; xq)61. Therefore, eG(x)¿dG(x; z) = dG(x; xq)+dG(xq; zs)+dG(zs; z)¿dG(u; xq)− 1 + dG(xq; zs) + dG(zs; z)¿dG(u; z) − 1 = diam(G) − 1, since dG(xq; zs) = dG(xq; zs) = 1. In this case, the second part ofthe theorem holds.
We then may assume that P2 has a chord joining some vertex xi to some vertex
zj with i6q − 1 and j6s. Then xq−1 and zs are connected in G − Lfz−1. Again, by Theorem 1 (4) and (2) and the chordality of G, dG(xq−1; zs)61. Thus, dG(y; x)6 dG(y; zs) + dG(zs; xq−1) + dG(xq−1; x)6dG(y; zs) + 1 + dG(xq−1; x) = dG(y; xq+1) + 1 + dG(xq−1; x)¡dG(y; xq+1) + dG(xq+1; xq−1) + dG(xq−1; x) = dG(y; x), a contradiction. This completes the proofofthe theorem.
Acknowledgements
The authors thank the referees for many constructive suggestions. References
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