Finding a central vertex in an HHD-free graph

In a graph G=(V; E), the eccentricity e(v) of a vertex v is max{d(v; u): u∈V}. The center of a graph is the set of vertices with minimum eccentricity. A house–hole–domino-free (HHD-free) graph is a graph which does not contain the house, the domino, and holes (cycles of length at least 2ve) as induced subgraphs. We present an algorithm which 2nds a central vertex of a HHD-free graph in O( 1:376|V |) time, where is the maximum degree of a vertex of G. Its complexity is linear in the case of weak bipolarizable graphs, chordal graphs, and distance-hereditary graphs. The algorithm uses special metric and convexity properties of HHD-free graphs. ? 2003 Elsevier B.V. All rights reserved.