Finding a central vertex in an HHD-free graph

Authors:
Victor Chepoi;Feodor Dragan
Affiliations:
Laboratoire d'Informatique Fondamentale, Université d'Aix Marseille II, Faculté des Sciences de Luminy, 163, Avenue de Luminy, 13288 Marseille Cedex 8, France;Department of Computer Science, Kent State University, Kent, OH 44242, USA
Venue:
Discrete Applied Mathematics
Year:
2003

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Abstract

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 five) as induced subgraphs. We present an algorithm which finds a central vertex of a HHD-free graph in O(@D^1^.^3^7^6|V|) time, where @D 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.