The Parameterized Complexity of Coloring Mixed Graphs

TL;DR

This paper studies the parameterized complexity of coloring mixed graphs, which combine directed and undirected edges, revealing new complexity results and algorithms based on structural parameters.

Contribution

It introduces and analyzes the parameterized complexity of mixed graph coloring, including new parameters like mixed neighborhood diversity, and establishes complexity boundaries.

Findings

Mixed coloring is W[1]-hard for treewidth.

Mixed coloring is paraNP-hard for neighborhood diversity.

Mixed coloring is fixed-parameter tractable when parameterized by mixed neighborhood diversity.

Abstract

A mixed graph contains (undirected) edges as well as (directed) arcs, thus generalizing undirected and directed graphs. A proper coloring $c$ of a mixed graph $G$ assigns a positive integer to each vertex such that $c(u)\neq c(v)$ for every edge $\{u,v\}$ and $c(u)<c(v)$ for every arc $(u,v)$ of $G$. As in classical coloring, the objective is to minimize the number of colors. Thus, mixed (graph) coloring generalizes classical coloring of undirected graphs and allows for more general applications, such as scheduling with precedence constraints, modeling metabolic pathways, and process management in operating systems; see a survey by Sotskov [Mathematics, 2020]. We initiate the systematic study of the parameterized complexity of mixed coloring. We focus on structural graph parameters that lie between cliquewidth and vertex cover, primarily with respect to the underlying undirected graph. Unlike classical coloring, which is fixed-parameter tractable (FPT) parameterized by treewidth or neighborhood diversity, we show that mixed coloring is W[1]-hard for treewidth and even paraNP-hard for neighborhood diversity. To utilize the directedness of arcs, we introduce and analyze natural generalizations of neighborhood diversity and cliquewidth to mixed graphs, and show that mixed coloring becomes FPT when parameterized by mixed neighborhood diversity. Further, we investigate how these parameters are affected if we add transitive arcs, which do not affect colorings. Finally, we provide tight bounds on the chromatic number of mixed graphs, generalizing known bounds on mixed interval graphs.