  Graph searching is one of the simplest and most widely used tools in graph
algorithms. Every graph search method is defined using some particular
selection rule, and the analysis of the corresponding vertex orderings can aid
greatly in devising algorithms, writing proofs of correctness, or recognition
of various graph families. We study graphs where the sets of vertex orderings
produced by two different search methods coincide. We characterise such graph
families for ten pairs from the best-known set of graph searches: Breadth First
Search (BFS), Depth First Search (DFS), Lexicographic Breadth First Search
(LexBFS) and Lexicographic Depth First Search (LexDFS), and Maximal
Neighborhood Search (MNS).
