Planar Open Rectangle-of-Influence Drawings with Non-aligned Frames

A straight-line drawing of a graph is an open weak rectangle-of-influence (RI) drawing, if there is no vertex in the relative interior of the axis-parallel rectangle induced by the end points of each edge. No algorithm is known to test whether a graph has a planar open weak RI-drawing, not even for inner triangulated graphs. In this paper, we study RI-drawings that must have a non-aligned frame, i.e., the graph obtained from removing the interior of every filled triangle is drawn such that no two vertices have the same coordinate. We give a polynomial algorithm to test whether an inner triangulated graph has a planar open weak RI-drawing with non-aligned frame.

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