The Parameterized Complexity of Oriented Colouring

The oriented colouring problem is intuitive and related to undirected colouring, yet remains NP-hard even on digraph classes with bounded traditional directed width measures. Recently we have also proved that it remains NP-hard in otherwise severely restricted digraph classes. However, unlike most other problems on directed graphs, the oriented colouring problem is not directly transferable to undirected graphs. In the article we look at the parameterized complexity of computing the oriented colouring of digraphs with bounded undirected width parameters, whereas the parameters "forget" about the orientations of arcs and parse them as edges. Specifically, we provide new complexity results for computing oriented colouring on digraphs of bounded undirected rank-width and a new algorithm for this problem on digraphs of bounded undirected tree-width.

[1]  Jaroslav Nesetril,et al.  Graphs and homomorphisms , 2004, Oxford lecture series in mathematics and its applications.

[2]  Robert Ganian,et al.  On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width , 2010, Discret. Appl. Math..

[3]  Éric Sopena,et al.  Oriented graph coloring , 2001, Discret. Math..

[4]  Marc Demange,et al.  Oriented Coloring: Complexity and Approximation , 2006, SOFSEM.

[5]  Robert Ganian,et al.  On Digraph Width Measures in Parameterized Algorithmics , 2009, IWPEC.

[6]  André Raspaud,et al.  Colored Homomorphisms of Colored Mixed Graphs , 2000, J. Comb. Theory, Ser. B.

[7]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[8]  Mamadou Moustapha Kanté.  The rank-width of Directed Graphs , 2007, ArXiv.

[9]  Robert Ganian,et al.  Better Polynomial Algorithms on Graphs of Bounded Rank-Width , 2009, IWOCA.

[10]  Glenn G. Chappell,et al.  Coloring with no 2-Colored P4's , 2004, Electron. J. Comb..

[11]  Fabian Wagner.  Hardness Results for Tournament Isomorphism and Automorphism , 2007, MFCS.

[12]  Paul D. Seymour,et al.  Graph minors. X. Obstructions to tree-decomposition , 1991, J. Comb. Theory, Ser. B.

[13]  B. Mohar,et al.  Graph Minors , 2009 .

[14]  Bruno Courcelle,et al.  The Monadic Second-order Logic of Graphs VI: On Several Representations of Graphs by Relational Structures , 1995, Discret. Appl. Math..

[15]  Sang-il Oum,et al.  Rank-width and vertex-minors , 2005, J. Comb. Theory, Ser. B.