  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. 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. Key words: Lexicographic breadth first
search, end-vertex, bipartite graphs, AT-free, proper interval bigraph,
characterization, algorithm.
