Transparent Rectangle Visibility Graphs

TL;DR

This paper introduces transparent rectangle visibility graphs (TRVGs), characterizes their properties for various graph classes, and establishes bounds and conditions for bipartite and torus TRVGs, expanding understanding of geometric graph representations.

Contribution

The paper characterizes TRVGs for several graph classes, determines maximum edges for bipartite TRVGs, and identifies conditions for complete bipartite and torus TRVGs, providing new insights into geometric graph representations.

Findings

Every threshold graph, tree, cycle, and certain grid graphs are TRVGs.

Maximum edges in bipartite TRVGs are 2n-2 for n vertices.

Complete bipartite graphs are TRVGs only in specific cases.

Abstract

A transparent rectangle visibility graph (TRVG) is a graph whose vertices can be represented by a collection of non-overlapping rectangles in the plane whose sides are parallel to the axes such that two vertices are adjacent if and only if there is a horizontal or vertical line intersecting the interiors of their rectangles. We show that every threshold graph, tree, cycle, rectangular grid graph, triangular grid graph and hexagonal grid graph is a TRVG. We also obtain a maximum number of edges of a bipartite TRVG and characterize complete bipartite TRVGs. More precisely, a bipartite TRVG with $n$ vertices has at most $2n-2$ edges. The complete bipartite graph $K_{p,q}$ is a TRVG if and only if $\min\{p,q\} \le 2$ or $(p,q) \in \{(3,3), (3,4)\}$. We prove similar results for the torus. Moreover, we study whether powers of cycles and their complements are TRVGs.