Strongly chordal digraphs and $\Gamma$-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 {\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.

[1]  Donald J. Rose,et al.  Toward Characterization of Perfect Elimination Digraphs , 1973, SIAM J. Comput..

[2]  Pavol Hell,et al.  Interval graphs, adjusted interval digraphs, and reflexive list homomorphisms , 2012, Discret. Appl. Math..

[3]  Stephan Kreutzer,et al.  Digraph measures: Kelly decompositions, games, and orderings , 2007, SODA '07.

[4]  Richard P. Anstee,et al.  Characterizations of Totally Balanced Matrices , 1984, J. Algorithms.

[5]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[6]  Thomas Britz,et al.  On free matrices , 2004 .

[7]  Pavol Hell,et al.  Bi-Arc Digraphs and Conservative Polymorphisms , 2016, ArXiv.

[8]  Jing Huang.  Representation characterizations of chordal bipartite graphs , 2006, J. Comb. Theory, Ser. B.

[9]  Anna Lubiw,et al.  Doubly lexical orderings of matrices , 1985, STOC '85.

[10]  Gerhard J. Woeginger,et al.  Permuting Matrices to Avoid Forbidden Submatrices , 1995, Discret. Appl. Math..

[11]  Gregory Gutin,et al.  Digraphs - Theory, Algorithms and Applications, Second Edition , 2009, Springer Monographs in Mathematics.

[12]  Douglas B. West,et al.  Interval digraphs: An analogue of interval graphs , 1989, J. Graph Theory.

[13]  Pavol Hell,et al.  Strict chordal and strict split digraphs , 2017, Discret. Appl. Math..

[14]  Pavol Hell,et al.  Interval-Like Graphs and Digraphs , 2018, MFCS.

[15]  Jan Arne Telle,et al.  Chordal Digraphs , 2009, WG.

[16]  Kunsoo Park,et al.  in Interval Graphs , 1997 .

[17]  G. Nemhauser,et al.  The k-Domination and k-Stability Problems on Sun-Free Chordal Graphs , 1984 .

[18]  Michael Drew Lamar,et al.  Split digraphs , 2010, Discret. Math..

[19]  A. Hoffman,et al.  Totally-Balanced and Greedy Matrices , 1985 .

[20]  Martin Charles Golumbic,et al.  Perfect Elimination and Chordal Bipartite Graphs , 1978, J. Graph Theory.

[21]  Martin Farber,et al.  Characterizations of strongly chordal graphs , 1983, Discret. Math..

[22]  D. J. KLEITMAN,et al.  A Note on Perfect Elimination Digraphs , 1974, SIAM J. Comput..