End-vertices of LBFS of (AT-free) bigraphs

摘要

Lexicographic Breadth First Search (LBFS) is one of fundamental graph searchalgorithms that has numerous applications, including recognition of graphclasses, computation of graph parameters, and detection of certain graphstructures. The well-known result of Rose, Tarjan and Lueker on theend-vertices of LBFS of chordal graphs has tempted researchers to study theend-vertices of LBFS of various classes of graphs, including chordal graphs,split graphs, interval graphs, and asteroidal triple-free (AT-free) graphs. Inthis paper we study the end-vertices of LBFS of bipartite graphs. We show thatdeciding whether a vertex of a bipartite graph is the end-vertex of an LBFS isan NP-complete problem. In contrast we characterize the end-vertices of LBFS ofAT-free bipartite graphs. Our characterization implies that the problem ofdeciding whether a vertex of an AT-free bipartite graph is the end-vertex of anLBFS is solvable in polynomial time. Key words: Lexicographic breadth firstsearch, end-vertex, bipartite graphs, AT-free, proper interval bigraph,characterization, algorithm.