Coloring Mixed and Directional Interval Graphs

TL;DR

This paper studies coloring rules for mixed and directional interval graphs, providing recognition algorithms, efficient chromatic number computation for directional cases, and proving NP-hardness for general mixed interval graphs.

Contribution

It introduces methods to recognize directional interval graphs and compute their chromatic number efficiently, and establishes NP-hardness for the general mixed interval graph coloring problem.

Findings

Recognition algorithm for directional interval graphs.

Efficient computation of chromatic number for directional interval graphs.

NP-hardness proof for coloring general mixed interval graphs.

Abstract

A mixed graph has a set of vertices, a set of undirected egdes, and a set of directed arcs. A proper coloring of a mixed graph $G$ is a function $c$ that assigns to each vertex in $G$ a positive integer such that, for each edge $uv$ in $G$, $c(u) \ne c(v)$ and, for each arc $uv$ in $G$, $c(u) < c(v)$. For a mixed graph $G$, the chromatic number $\chi(G)$ is the smallest number of colors in any proper coloring of $G$. A directional interval graph is a mixed graph whose vertices correspond to intervals on the real line. Such a graph has an edge between every two intervals where one is contained in the other and an arc between every two overlapping intervals, directed towards the interval that starts and ends to the right. Coloring such graphs has applications in routing edges in layered orthogonal graph drawing according to the Sugiyama framework; the colors correspond to the tracks for routing the edges. We show how to recognize directional interval graphs, and how to compute their chromatic number efficiently. On the other hand, for mixed interval graphs, i.e., graphs where two intersecting intervals can be connected by an edge or by an arc in either direction arbitrarily, we prove that computing the chromatic number is NP-hard.