AI 中文总结
介绍角矩形可见性图（CRVGs）及其特殊类型南和西南CRVGs，证明它们边数的上限并表明边界是紧的，还证明了\(n\)个顶点CRVGs边数范围，最后将几个图族分类为相应类型。
AI 中文摘要
我们引入角矩形可见性图（CRVGs），它是由两类几何定义的图：矩形可见性图（RVGs）和影响矩形图（RIGs）组合而成。CRVG的顶点由平面中的轴平行矩形表示，边由轴平行矩形表示，其一个角在顶点矩形的角上，对角在另一个顶点矩形的边界上，且内部没有顶点矩形。我们还考虑了只在一个或两个方向可见的CRVGs（南CRVGs和西南CRVGs）。我们证明南CRVGs最多有\(\left[\frac{n^2}{4}\right]+n - 2\)条边且此边界是紧的，西南CRVGs最多有\(\left[\frac{n^2}{3}+\frac{n}{3}\right]-1\)条边且此边界是紧的。我们证明\(n\)个顶点的CRVGs最多有\(e\)条边，其中\(\lfloor \frac{3n^2}{8} \rfloor \leq e \leq \lfloor \frac{2n^2}{5} \rfloor\)。最后，我们将几个图族分类为CRVGs、SCRVGs和SWCRVGs。
英文摘要
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 $\lfloor \frac{3n^2}{8} \rfloor \leq e \leq \lfloor \frac{2n^2}{5} \rfloor$. Finally, we classify several families of graphs as CRVGs, SCRVGs, and SWCRVGs.
