On the power of standard DFS and BFS

2026-05-04

AI Summary

This study addresses the efficient recognition and certification of specific graph classes using standard graph search algorithms. The authors propose novel linear-time methods that require only one or two traversals: a simplified depth-first search (DFS) suffices to recognize trivially perfect graphs, a single breadth-first search (BFS) identifies split graphs and bipartite chain graphs, and the recognition of proper interval graphs is improved from three LexBFS sweeps to just two BFS traversals. Furthermore, the work establishes that the canonical ordering characterizing proper interval graphs is unique up to reversal and true-twin permutations. These results significantly streamline existing recognition algorithms and provide a clearer structural characterization alongside an efficient certification framework.

Abstract

It is well-known since the seventies of last century that Depth First Search (DFS) can be used to compute strongly connected components [RE. Tarjan. SIAM Journal on Computing, 1972] and Breadth First Search (BFS) can be used to compute distance in graphs [GY. Handler. Transportation Science, 1973]. We furthermore demonstrate that these standard graph searches are powerful enough to recognize and certify several well-structured graph classes. Specifically, we provide a single DFS approach for recognizing and certifying trivially perfect graphs that is significantly simpler than previous methods using [FPM. Chu. Information Processing Letters, 2008]. We further show that a single BFS can recognize split graphs and bipartite chain graphs, and we improve upon the triple LexBFS algorithm for proper interval graphs [DG. Corneil. Discrete Applied Mathematics, 2004] by proposing a two consecutive BFS recognition scheme. These results are underpinned by characterizations using vertex orderings that avoid specific patterns [L. Feuilloley, M. Habib. SIAM Journal on Discrete Mathematics, 2021]. Finally, we provide a structural study of connected proper interval graphs, proving that their characterizations via special orderings are unique up to reversal and the permutation of true twins.

Problem

• graph recognition
• DFS
• BFS
• graph classes
• vertex orderings

Innovation

• DFS
• BFS
• graph recognition
• vertex ordering
• proper interval graphs