Mutual Witness Gabriel Drawings of Complete Bipartite Graphs

William J. Lenhart; Giuseppe Liotta

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arXiv · 2209.01004

Mutual Witness Gabriel Drawings of Complete Bipartite Graphs

William J. Lenhart·Giuseppe Liotta

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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.

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William J. Lenhart, Giuseppe Liotta. 2022-09-02. Mutual Witness Gabriel Drawings of Complete Bipartite Graphs. https://arxiv.org/abs/2209.01004

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