Corner Rectangle Visibility Graphs

Abstract

We introduce corner rectangle visibility graphs (CRVGs), a combination of two geometrically defined classes of graphs: rectangle visibility graphs (RVGs) and rectangle-of-influence graphs (RIGs). A CRVG has vertices represented by axis-parallel rectangles in the plane, and edges represented by axis-parallel rectangles with one corner at a corner of a vertex-rectangle, an opposite corner at the boundary of another vertex-rectangle, and no vertex-rectangles in their interiors. We also consider CRVGs that only see in one or two directions (south CRVGs and southwest CRVGs). We prove that south CRVGs have at most $\left[\frac{n^2}{4}\right]+n-2$ edges, and this bound is tight. This is the same as the tight edge bound for closed RIGs, but they are different graph classes. We also show that southwest CRVGs have at most $\left[\frac{n^2}{3}+\frac{n}{3}\right]-1$ edges, and this bound is tight. We prove that CRVGs on $n$ vertices have at most $e$ edges, where $\lfloor \frac{3n^2}{8}\rfloor \le e\le \lfloor \frac{2n^2}{5}\rfloor$. Finally, we classify several families of graphs as CRVGs, SCRVGs, and SWCRVGs.

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