Graphs where Search Methods are Indistinguishable

TL;DR

This paper characterizes classes of graphs where different search algorithms produce identical vertex orderings, enhancing understanding of their relationships and aiding algorithm design.

Contribution

It provides a comprehensive characterization of graph families where ten pairs of common search methods yield indistinguishable vertex orderings.

Findings

Characterization of graph families for BFS and DFS

Characterization of graph families for LexBFS and LexDFS

Characterization of graph families for MNS and other searches

Abstract

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).