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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.