Mutual Witness Gabriel Drawings of Complete Bipartite Graphs

TL;DR

This paper characterizes pairs of complete bipartite graphs that can be represented as mutual witness Gabriel drawings, providing a linear time testing algorithm and showing limitations for more complex multipartite graphs.

Contribution

It offers a complete characterization and efficient testing algorithm for mutual witness Gabriel drawings of complete bipartite graphs, and identifies cases where such drawings are impossible.

Findings

Characterization of bipartite graph pairs admitting mutual witness Gabriel drawings

Development of a linear time testing algorithm

Proof that certain multipartite graphs do not admit such drawings

Abstract

Let $\Gamma$ be a straight-line drawing of a graph and let $u$ and $v$ be two vertices of $\Gamma$. The Gabriel disk of $u,v$ is the disk having $u$ and $v$ as antipodal points. A pair $\langle \Gamma_0,\Gamma_1 \rangle$ of vertex-disjoint straight-line drawings form a mutual witness Gabriel drawing when, for $i=0,1$, any two vertices $u$ and $v$ of $\Gamma_i$ are adjacent if and only if their Gabriel disk does not contain any vertex of $\Gamma_{1-i}$. We characterize the pairs $\langle G_0,G_1 \rangle $ of complete bipartite graphs that admit a mutual witness Gabriel drawing. The characterization leads to a linear time testing algorithm. We also show that when at least one of the graphs in the pair $\langle G_0, G_1 \rangle $ is complete $k$-partite with $k>2$ and all partition sets in the two graphs have size greater than one, the pair does not admit a mutual witness Gabriel drawing.