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{ω(G), λ(G)}-approximation algorithm, where ω(G) is the size of a largest clique and λ(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.

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An obstacle representation of a graph G consists of a set of polygonal obstacles and a drawing of G as a visibility graph with respect to the obstacles: vertices are mapped to points and edges to straight-line segments such that each edge avoids all obstacles whereas each non-edge intersects at least one obstacle. Obstacle representations have been investigated quite intensely over the last few years. Here we focus on outside-obstacle representations that use only one obstacle in the outer face of the drawing. It is known that every outerplanar graph admits such a representation [Alpert, Koch, Laison; DCG 2010]. We strengthen this result by showing that every partial 2-tree has an outside-obstacle representation. We also consider a restricted version of outside-obstacle representations where the vertices lie on a regular polygon. We construct such regular representations for partial outerpaths, partial cactus graphs, and partial grids.