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

A straight-line drawing of a plane graph is called an open rectangle-ofinfluence 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 necessarily 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/logn)-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.