On the power of standard DFS and BFS

Binh-Minh Bui-Xuan; Michel Habib; Fabien de Montgolfier; Renaud Torfs

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arXiv · 2605.02387

On the power of standard DFS and BFS

Binh-Minh Bui-Xuan·Michel Habib·Fabien de Montgolfier·Renaud Torfs

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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.

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Binh-Minh Bui-Xuan, Michel Habib, Fabien de Montgolfier, Renaud Torfs. 2026-05-04. On the power of standard DFS and BFS. https://arxiv.org/abs/2605.02387

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