Strongly chordal digraphs and $\Gamma$-free matrices

TL;DR

This paper introduces strongly chordal digraphs, explores their properties, and provides polynomial-time recognition algorithms and characterizations for specific subclasses, extending existing graph classes.

Contribution

It defines strongly chordal digraphs, relates them to matrix avoidance properties, and offers recognition algorithms and characterizations for symmetric, tournament, and balanced cases.

Findings

Polynomial-time recognition algorithms for certain subclasses.

Forbidden subgraph characterizations established.

Extended algorithms for minimum dominating set problem.

Abstract

We define strongly chordal digraphs, which generalize strongly chordal graphs and chordal bipartite graphs, and are included in the class of chordal digraphs. They correspond to square 0,1 matrices that admit a simultaneous row and column permutation avoiding the {\Gamma} matrix. In general, it is not clear if these digraphs can be recognized in polynomial time, and we focus on symmetric digraphs (i.e., graphs with possible loops), tournaments with possible loops, and balanced digraphs. In each of these cases we give a polynomial-time recognition algorithm and a forbidden induced subgraph characterization. We also discuss an algorithm for minimum general dominating set in strongly chordal graphs with possible loops, extending and unifying similar algorithms for strongly chordal graphs and chordal bipartite graphs.