Many graph search algorithms use a vertex labeling to compute an ordering of the vertices. We examine such algorithms which compute a peo (perfect elimination ordering) of a chordal graph, and corresponding algorithms which compute an meo (minimal elimination ordering) of a non-chordal graph, an ordering used to compute a minimal triangulation of the input graph. \par We express all known peo-computing search algorithms as instances of a generic algorithm called MLS (Maximal Label Search) and generalize Algorithm MLS into CompMLS, which can compute any peo. \par We then extend these algorithms to versions which compute an meo, and likewise generalize all known meo-computing search algorithms. We show that not all minimal triangulations can be computed by such a graph search, and, more surprisingly, that all these search algorithms compute the same set of minimal triangulations, even though the computed meos are different. \par Finally, we present a complexity analysis of these algorithms.