On the power of standard DFS and BFS

TL;DR

This paper demonstrates the power of standard DFS and BFS algorithms in recognizing and certifying various well-structured graph classes, simplifying existing methods and improving recognition schemes.

Contribution

It introduces simplified DFS and BFS-based algorithms for recognizing complex graph classes and enhances existing recognition schemes with more efficient BFS-based methods.

Findings

A single DFS approach recognizes trivially perfect graphs more simply.

A single BFS recognizes split and bipartite chain graphs.

A two-step BFS scheme improves recognition of proper interval graphs.

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.