Mutual witness Gabriel drawings of complete bipartite graphs

Abstract

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 〈Γ0,Γ1〉 of vertex-disjoint straight-line drawings forms a mutual witness Gabriel drawing when, for i=0,1, any two vertices u and v of Γi are adjacent if and only if their Gabriel disk does not contain any vertex of Γ1−i. We characterize the pairs 〈G0,G1〉 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 the pair 〈G0,G1〉 consists of two complete multi-partite graphs whose partition sets all have size greater than one, then the pair does not admit a mutual witness Gabriel drawing unless the pair is 〈K2,2,K2,2〉.