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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.

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