The Parameterized Complexity of Coloring Mixed Graphs

2026-04-16

AI Summary

This study addresses the parameterized complexity of the mixed graph coloring problem, which involves assigning colors to vertices of a graph containing both undirected edges and directed arcs such that adjacent vertices receive distinct colors and the color of the tail of every arc is strictly less than that of its head. The work presents the first systematic application of parameterized complexity theory to this problem, introducing novel structural parameters such as mixed neighborhood diversity. Through W[1]-hardness and paraNP-hardness reductions, it demonstrates the limitations of classical parameters like treewidth and neighborhood diversity: the problem is W[1]-hard when parameterized by treewidth and paraNP-hard under neighborhood diversity. In contrast, it becomes fixed-parameter tractable when parameterized by mixed neighborhood diversity. The paper also establishes tight upper and lower bounds on the chromatic number for mixed graphs.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Social Networks and Social Media: Social mining and social search on the Web

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)!=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.

Problem

• mixed graph coloring
• parameterized complexity
• structural graph parameters
• chromatic number
• precedence constraints

Innovation

• mixed graph coloring
• parameterized complexity
• mixed neighborhood diversity
• fixed-parameter tractability
• structural graph parameters