A \emph{mixed interval graph} is an interval graph that has, for every pair of intersecting intervals, either an arc (directed arbitrarily) or an (undirected) edge. We are particularly interested in scenarios where edges and arcs are defined by the geometry of intervals. In a proper coloring of a mixed interval graph $G$, an interval $u$ receives a lower (different) color than an interval $v$ if $G$ contains arc $(u,v)$ (edge $\{u,v\}$). Coloring of mixed graphs has applications, for example, in scheduling with precedence constraints; see a survey by Sotskov [Mathematics, 2020]. For coloring general mixed interval graphs, we present a $\min \{\omega(G), \lambda(G)+1 \}$-approximation algorithm, where $\omega(G)$ is the size of a largest clique and $\lambda(G)$ is the length of a longest directed path in $G$. For the subclass of \emph{bidirectional interval graphs} (introduced recently for an application in graph drawing), we show that optimal coloring is NP-hard. This was known for general mixed interval graphs. We introduce a new natural class of mixed interval graphs, which we call \emph{containment interval graphs}. In such a graph, there is an arc $(u,v)$ if interval $u$ contains interval $v$, and there is an edge $\{u,v\}$ if $u$ and $v$ overlap. We show that these graphs can be recognized in polynomial time, that coloring them with the minimum number of colors is NP-hard, and that there is a 2-approximation algorithm for coloring.

翻译：混合区间图是一类区间图，对于每对相交的区间，该图包含一条（任意定向的）有向弧或一条（无向）边。我们特别关注由区间几何性质定义的边和弧的情形。在混合区间图$G$的正常着色中，若$G$包含弧$(u,v)$（或边$\{u,v\}$），则区间$u$获得的颜色需低于（或不同于）区间$v$的颜色。混合图的着色在具有优先约束的调度问题中具有应用（参见Sotskov的综述[Mathematics, 2020]）。针对一般混合区间图的着色问题，我们提出了一种$\min \{\omega(G), \lambda(G)+1 \}$-近似算法，其中$\omega(G)$是最大团的规模，$\lambda(G)$是$G$中最长有向路径的长度。对于最近因图绘制应用而引入的\emph{双向区间图}子类，我们证明其最优着色问题是NP难的（这一问题此前已知对一般混合区间图成立）。我们引入了一类新的混合区间图——\emph{包含区间图}：若区间$u$包含区间$v$，则存在弧$(u,v)$；若$u$与$v$部分重叠，则存在边$\{u,v\}$。我们证明此类图可在多项式时间内识别，其最小着色问题是NP难的，并给出了一个2-近似着色算法。