Mutual Witness Gabriel Drawings of Complete Bipartite Graphs
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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 ⟨Γ_0,Γ_1 ⟩ 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 ⟨ G_0,G_1 ⟩ 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 ⟨ G_0, G_1 ⟩ 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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