A 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 interested in mixed interval graphs where the type of connection of two vertices is determined by geometry. 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}). We introduce a new natural class of mixed interval graphs, which we call 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. For coloring general mixed interval graphs, we present a min{{\omega}(G), {\lambda}(G)}-approximation algorithm, where {\omega}(G) is the size of a largest clique and {\lambda}(G) is the length of a longest induced directed path in G. For the subclass of bidirectional interval graphs (introduced recently), we show that optimal coloring is NP-hard.

翻译：混合区间图是一种区间图，其中每对相交区间要么具有（任意定向的）有向弧，要么具有（无向的）边。我们关注连接类型由几何结构决定的混合区间图。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获得的颜色比区间v更低（或不同）。我们引入了一类新的自然混合区间图，称为包含区间图。在此类图中，若区间u包含区间v，则存在弧(u,v)；若u与v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，但最小颜色数着色问题是NP难的，且存在2-近似着色算法。针对一般混合区间图的着色问题，我们提出了最小{ω(G), λ(G)}-近似算法，其中ω(G)是最大团的大小，λ(G)是G中最长诱导有向路径的长度。对于最近引入的双向区间图子类，我们证明其最优着色问题是NP难的。