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Mutual Witness Gabriel Drawings of Complete Bipartite Graphs

Let $Γ$ be a straight-line drawing of a graph and let $u$ and $v$ be two vertices of $Γ$. The Gabriel disk of $u,v$ is the disk having $u$ and $v$ as antipodal points. A pair $\langle Γ_0,Γ_1 \rangle$ of vertex-disjoint straight-line drawings form a mutual witness Gabriel drawing when, for $i=0,1$, any two vertices $u$ and $v$ of $Γ_i$ are adjacent if and only if their Gabriel disk does not contain any vertex of $Γ_{1-i}$. 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.