Coloring Mixed and Directional Interval Graphs

TL;DR

This work studies coloring problems on mixed and directional interval graphs, motivated by edge routing in layered Sugiyama drawings. It shows that a left-to-right greedy coloring yields an optimal coloring for directional interval graphs and provides a linearithmic algorithm, while demonstrating NP-hardness for mixed interval graphs and their proper variants through a 3-SAT reduction. The paper also develops a quadratic-time recognition algorithm for directional interval graphs using MPQ-trees and two-dimensional posets, enabling construction of directional representations when feasible. Collectively, these results delineate the boundary between tractable and intractable coloring in interval-graph-based models and link geometric representations with graph-theoretic colorings relevant to practical routing problems.

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 $χ(G)$ 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.

Figures (4)

• Figure 1: Separate greedy assignment of left-going and right-going edges to tracks.
• Figure 2: A staircase and its intermediate intervals, which form a clique in $U(G^+)$.
• Figure 3: A variable gadget for a variable $v_i$.
• Figure 4: A clause gadget for a clause $c_j = v_i \lor \lnot v_k \lor v_\ell$, where $z \notin \left\{i,k,\ell\right\}$.

Theorems & Definitions (18)

• lemma thmcounterlemma: \ref{['lem:runtimegreedy*']}
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary: \ref{['cor:approx*']}
• theorem thmcountertheorem
• proof
• corollary thmcountercorollary: \ref{['clm:hardnesspropper*']}
• theorem thmcountertheorem: \ref{['thm:recognition*']}
• lemma thmcounterlemma: Rotating PQ-trees
• proof