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

Lexicographic Breadth First Search (LBFS) is one of fundamental graph search algorithms that has numerous applications, including recognition of graph classes, computation of graph parameters, and detection of certain graph structures. The well-known result of Rose, Tarjan and Lueker on the end-vertices of LBFS of chordal graphs has tempted researchers to study the end-vertices of LBFS of various classes of graphs, including chordal graphs, split graphs, interval graphs, and asteroidal triple-free (AT-free) graphs. In this paper we study the end-vertices of LBFS of bipartite graphs. We show that deciding whether a vertex of a bipartite graph is the end-vertex of an LBFS is an NP-complete problem. In contrast we characterize the end-vertices of LBFS of AT-free bipartite graphs (or equivalently, proper interval bigraphs). Our characterization implies that the problem of deciding whether a vertex of an AT-free bipartite graph is the end-vertex of an LBFS is solvable in polynomial time.