Multipartite and Structural Results on Transparent Rectangle Visibility Graphs

TL;DR

This paper studies the properties of transparent rectangle visibility graphs (TRVGs), classifies certain types, proves non-existence of specific graphs, and introduces a new related graph class, expanding understanding of geometric graph representations.

Contribution

It classifies complete k-partite TRVGs, proves non-existence of certain graphs as TRVGs, and introduces the intersecting TRVG, advancing geometric graph theory.

Findings

$K_{3,3,3}$ is not a TRVG

Complete $k$-partite TRVGs are classified

Existence of graphs that are ITRVG but not TRVG

Abstract

We consider a graph representation in the plane, called the transparent rectangle visibility graph (TRVG), where each vertex is represented by a rectangle in the plane with sides parallel to the plane axes, in a way that any two vertices are adjacent if and only if a vertical or horizontal line can be drawn from the interior of one rectangle to the other. Expanding upon previously done work by Juntarapomdach and Kittipassorn, we show that $K_{3,3,3}$ is not a TRVG, and classify complete $k$-partite TRVGs. We also prove that the complement of $C^2_n$ is not a TRVG whenever $n \geq 15$, and that every $k$-partite TRVG with $n$ vertices has at most $2(k-1)n-k(k-1)$ edges. Furthermore, we introduce a novel representation, the intersecting transparent rectangle visibility graph (ITRVG), and show that there exists a graph that is an ITRVG but not a TRVG.