CENTERS OF CHORDAL GRAPHS

Abstract

In a graph G = (V, E), the eccentricity e(S) of a subset S is contained in or equal to V is max(x is-an-element-of V) min(y is-an-element-of S) d(x, y); and e(x) stands for e({x}). The diameter of G is max(x is-an-element-of V) e(x), the radius r(G) of G is min(x is-an-element-of V) e(x) and the clique radius cr(G) is min e(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) less-than-or-equal-to d(G) less-than-or-equal-to 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.