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arXiv · 2509.18541

Strong chordality in tournaments and multipartite tournaments with possible loops

Pavol Hell·César Hernández-Cruz·Jing Huang
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Abstract

Strongly chordal digraphs are included in the class of chordal digraphs and generalize strongly chordal graphs and chordal bipartite graphs. They are the digraphs that admit a linear ordering of its vertex set for which their adjacency matrix does not contain the $Γ$ matrix as a submatrix. In general, it is not clear if these digraphs can be recognized in polynomial time. We focus on multipartite tournaments with possible loops. We give a polynomial-time recognition algorithm and a forbidden induced subgraph characterization of the strong chordality for each of the following cases: tournaments with possible loops, reflexive multipartite tournaments, irreflexive bipartite tournaments, irreflexive tournaments minus one arc, and balanced digraphs. In addition, we prove that in a strongly chordal digraph the minimum size of a total dominating set equals the maximum number of disjoint in-neighborhoods, and this number can be calculated in linear time given a $Γ$-free ordering of the input graph.

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Pavol Hell, César Hernández-Cruz, Jing Huang. 2025-09-23. Strong chordality in tournaments and multipartite tournaments with possible loops. https://arxiv.org/abs/2509.18541

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