Colourings, homomorphisms, and partitions of transitive digraphs

Abstract We investigate the complexity of generalizations of colourings (acyclic colourings, ( k , l ) -colourings, homomorphisms, and matrix partitions), for inputs restricted to the class of transitive digraphs. Even though transitive digraphs are nicely structured, many problems are intractable, and their complexity turns out to be difficult to classify. We present some motivational results and several open problems.

[1]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[2]  Noga Alon,et al.  Maximum directed cuts in acyclic digraphs , 2007 .

[3]  Pavol Hell,et al.  On realizations of point determining graphs, and obstructions to full homomorphisms , 2008, Discret. Math..

[4]  Victor Neumann-Lara,et al.  The dichromatic number of a digraph , 1982, J. Comb. Theory, Ser. B.

[5]  Jian Song,et al.  A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs , 2014, J. Graph Theory.

[6]  Pavol Hell,et al.  Digraph matrix partitions and trigraph homomorphisms , 2006, Discret. Appl. Math..

[7]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[8]  Jørgen Bang-Jensen,et al.  Quasi-transitive digraphs , 1995, J. Graph Theory.

[9]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[10]  M. Dinneen,et al.  Minor-order obstructions for the graphs of vertex cover 6 , 2002 .

[11]  Michael J. Dinneen,et al.  Properties of vertex cover obstructions , 2007, Discret. Math..

[12]  John Gimbel,et al.  Partitioning graphs into complete and empty graphs , 2009, Discret. Math..

[13]  M. Chudnovsky.  Berge trigraphs , 2006 .

[14]  Pavol Hell,et al.  Graph partitions with prescribed patterns , 2014, Eur. J. Comb..

[15]  Gary MacGillivray,et al.  The complexity of colouring by locally semicomplete digraphs , 2010, Discret. Math..

[16]  Sulamita Klein,et al.  List Partitions , 2003, SIAM J. Discret. Math..

[17]  Pavol Hell,et al.  Matrix partitions of perfect graphs , 2006, Discret. Math..

[18]  Bruce A. Reed,et al.  Digraph Girth via Chromatic Number , 2013, SIAM J. Discret. Math..

[19]  Hortensia Galeana-Sánchez,et al.  K-kernels in Generalizations of Transitive Digraphs , 2011, Discuss. Math. Graph Theory.

[20]  Gregory Gutin,et al.  Digraphs - theory, algorithms and applications , 2002 .