Influence of the tie-break rule on the end-vertex problem

End-vertices of a given graph search may have some nice properties (as for example it is well known that the last vertex of Lexicographic Breadth First Search (LBFS) in a chordal graph is simplicial, see [16]). Therefore it is interesting to consider if these vertices can be recognized in polynomial time or not, as first studied in [6]. A graph search is a mechanism for systematically visiting the vertices of a graph. At each step of a graph search, the key point is the choice of the next vertex to be explored. Graph searches only differ by this selection mechanism during which a tie-break rule is used. In this paper we study how the choice of the tie-break in case of equality during the search, for a given graph search including the classic ones such as BFS and DFS, can determine the complexity of the end-vertex problem. In particular we prove a counterintuitive NPcompleteness result for Breadth First Search solving a problem raised in [6].