End-Vertices of AT-free Bigraphs

Abstract

The end-vertex problem for a search algorithm asks whether a vertex of the input graph is the last visited vertex of an execution of that search algorithm. We consider the end-vertex problem restricted to AT-free bigraphs for various search algorithms: Breadth-First Search (BFS), Lexicographic Breadth-First Search (LBFS), Depth-First Search (DFS), and Maximal Neighbourhood Search (MNS). Deciding whether a vertex of a graph is the end-vertex of any of these search algorithms is NP-complete in general. We show that we can decide whether a vertex is an end-vertex of BFS or MNS in polynomial time on AT-free bigraphs. Additionally, we show that we can decide whether a vertex is an end-vertex of DFS or LBFS in linear time on AT-free bigraphs; this improves the LBFS end-vertex complexity on this class of graphs.