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.

翻译：自上世纪七十年代以来，深度优先搜索（DFS）被用于计算强连通分量 [RE. Tarjan. SIAM Journal on Computing, 1972]，广度优先搜索（BFS）被用于计算图中的距离 [GY. Handler. Transportation Science, 1973]，这一认识已广为人知。我们进一步证明，这些标准图搜索方法足以识别并认证若干结构良好的图类。具体而言，我们提出一种单一DFS方法，用于识别和认证平凡完美图，其流程较[FPM. Chu. Information Processing Letters, 2008]提出的先前方法显著简化。此外，我们证明单一BFS即可识别分裂图与二分链图，并通过提出一种连续两次BFS的识别方案，改进了针对真区间图的三重LexBFS算法 [DG. Corneil. Discrete Applied Mathematics, 2004]。这些成果基于利用避免特定模式的顶点序进行图类刻画的原理 [L. Feuilloley, M. Habib. SIAM Journal on Discrete Mathematics, 2021]。最后，我们对连通真区间图的结构展开研究，证明其通过特殊顶点序的刻画在反转对称与真孪顶点置换下具有唯一性。