Abstract
Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G = ( V , A ) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows that finding the chromatic number of an oriented graph is an NP-hard problem, too. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.
Highlights
Graph coloring is one of the basic problems in graph theory, which has already been considered in the 19th century
It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one
We have considered vertex coloring on oriented graphs
Summary
Graph coloring is one of the basic problems in graph theory, which has already been considered in the 19th century. For several special undirected graph classes the oriented chromatic number has been bounded. Among these are outerplanar graphs [6], planar graphs [7], and Halin graphs [8]. In [9], Ganian has shown an FPT-algorithm for OCN w.r.t. the parameter tree-width (of the underlying undirected graph) He has shown that OCN is DET-hard (DET is the class of decision problems which are reducible in logarithmic space to the problem of computing the determinant of an integer valued n × n-matrix.) for classes of oriented graphs, such that the underlying undirected class has bounded rank-width. Since oriented co-graphs have a directed NLC-width of one [12], our results provide a useful basis for exploring the complexity of OCN related to width parameters (cf. Section 7)
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