Multipartite and Structural Results on Transparent Rectangle Visibility Graphs

TL;DR

The paper investigates transparent rectangle visibility graphs (TRVGs), extending rectangle visibility concepts by allowing transparency and intersections, and builds on CJTK's bipartite classifications to explore complete multipartite and cyclic-square complements. Its main approaches combine geometric constructions (bounding boxes, strip arguments) with induced-subgraph techniques to derive non-TRVG results, as well as generalized edge-count bounds for multipartite TRVGs. Key contributions include proving $K_{3,3,3}$ is not a TRVG, fully classifying complete $k$-partite TRVGs, showing $D_n^2$ is not a TRVG for $n\ge15$, establishing the edge bound $e(G)\le 2(k-1)n-k(k-1)$, and introducing the broader ITRVG class with a demonstrable example strictly outside TRVG. These results advance the understanding of visibility representations and provide a framework for future studies in discrete geometry and VLSI-inspired applications.

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.

Figures (9)

• Figure 1: A difference between a TRVG and an RVG
• Figure 2: Possible rectangle representations of points in the first part
• Figure 3: Examples of placement of red and yellow rectangles in (a) Case 1.1 and (b) Case 1.2
• Figure 4: Examples of placement of red rectangles in Case 2
• Figure 5: (a) A representation of $K_{1,a_r,...,a_k}$ and (b) a representation of $K_{2,a_r,...,a_k}$ constructed from a given representation of $K_{a_r,...,a_k}$

Theorems & Definitions (15)

• Theorem 1
• Lemma 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 5
• Theorem 6
• proof