Open Rectangle-of-Influence Drawings of Inner Triangulated Plane Graphs

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A straight-line drawing of a plane graph is called an open rectangle-of-influence drawing if there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of every edge. In an inner triangulated plane graph, every inner face is a triangle although the outer face is not always a triangle. In this paper, we first obtain a sufficient condition for an inner triangulated plane graph G to have an open rectangle-of-influence drawing; the condition is expressed in terms of a labeling of angles of a subgraph of G. We then present an O(n1.5/log n)-time algorithm to examine whether G satisfies the condition and, if so, construct an open rectangle-of-influence drawing of G on an (n-1)×(n-1) integer grid, where n is the number of vertices in G.