Strongly chordal digraphs and Γ-free matrices

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 Γ 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. 1 Background and definitions A number of interesting graph classes have been extended to digraphs, including interval graphs [8], chordal graphs [11,13,25], split graphs [13,22], and graphs of bounded treewidth [18,19]. In most cases, there is more than one way to define such a generalization, and it is not obvious which one best captures the analogy to the undirected case. (In the undirected case there may be several equivalent characterizations of the graphs in the class, and each may suggest a different ∗The authors gratefully acknowledge support from NSERC Canada †pavol@sfu.ca ‡cesar@cs.cinvestav.mx §huangj@uvic.ca ¶chlin@math.nsysu.edu.tw 1 ar X iv :1 90 9. 03 59 7v 2 [ m at h. C O ] 1 3 N ov 2 01 9 generalization, which are not equivalent in the context of digraphs.) It seems to be the case that often the most successful generalizations use the ordering characterization of the undirected concept, or, equivalently, its characterization by forbidden submatrices of the adjacency matrix. Consider first the undirected notion of an interval graph. Since every interval intersects itself, we will assume each vertex has a loop. Then interval graphs are known to have the following ordering characterization [8]. (There are other ordering characterizations of interval graphs, but this one turns out to be most useful; however, it only applies if every vertex is considered adjacent to itself.) A graph G is an interval graph if and only if its vertices can be ordered as v1, v2, . . . , vn so that if i < j and k < `, not necessarily all distinct, then for viv` ∈ E(G), vjvk ∈ E(G) we also have vjv` ∈ E(G). Equivalently, G is an interval graph if and only if the rows and columns of its adjacency matrix can be simultaneously permuted to avoid a submatrix of the form [ ∗ 1 1 0 ] where ∗ can be either 0 or 1. In [8], the authors analogously define a digraph analogue of interval graphs as follows. A digraph with a loop at every vertex is an adjusted interval digraph if the rows and columns of its adjacency matrix can be simultaneously permuted to avoid a submatrix of the form [ ∗ 1 1 0 ]