Abstract
AbstractLet \(\varGamma \) be a straight-line drawing of a graph and let u and v be two vertices of \(\varGamma \). The Gabriel disk of u, v is the disk having u and v as antipodal points. A pair \(\langle \varGamma _0,\varGamma _1 \rangle \) of vertex-disjoint straight-line drawings form a mutual witness Gabriel drawing when, for \(i=0,1\), any two vertices u and v of \(\varGamma _i\) are adjacent if and only if their Gabriel disk does not contain any vertex of \(\varGamma _{1-i}\). 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.KeywordsProximity drawingsGabriel drawingsWitness proximity drawingsSimultaneous drawing of two graphs
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