Given a graph G=(V,E), a vertex ordering of G is a total order v1,v2,...,vn of V. A graph search algorithm is a systematic method for visiting each vertex in a graph, naturally producing a vertex ordering of the graph. We explore the problem of determining whether a given vertex in a graph can be the end (last) vertex of a search ordering for various common graph search algorithms when restricted to various graph classes, as well as the related problem of determining if a vertex is an end-vertex when a start vertex is specified for the search. The former is referred to as the end-vertex problem, and the latter is the beginning-end-vertex problem. For the beginning-end-vertex problem, we show it is NP-complete on bipartite graphs as well as degree restricted bipartite graphs for Lexicographic Breadth First Search, but solvable in polynomial time on split graphs for Breadth First Search. We show that the end-vertex problem is tractable for Lexicographic Breadth First Search on proper interval bigraphs and for Lexicographic Depth First Search on chordal graphs. Further, we show that the problem is NP-complete for Lexicographic Breadth First Search and Depth First Search on bipartite graphs. / Graduate