Mutual Witness Gabriel Drawings of Complete Bipartite Graphs
William J. Lenhart, Giuseppe Liotta

TL;DR
This paper characterizes pairs of complete bipartite graphs that can be represented as mutual witness Gabriel drawings, providing a linear time testing algorithm and showing limitations for more complex multipartite graphs.
Contribution
It offers a complete characterization and efficient testing algorithm for mutual witness Gabriel drawings of complete bipartite graphs, and identifies cases where such drawings are impossible.
Findings
Characterization of bipartite graph pairs admitting mutual witness Gabriel drawings
Development of a linear time testing algorithm
Proof that certain multipartite graphs do not admit such drawings
Abstract
Let be a straight-line drawing of a graph and let and be two vertices of . The Gabriel disk of is the disk having and as antipodal points. A pair of vertex-disjoint straight-line drawings form a mutual witness Gabriel drawing when, for , any two vertices and of are adjacent if and only if their Gabriel disk does not contain any vertex of . We characterize the pairs 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 is complete -partite with 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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Taxonomy
TopicsComputational Geometry and Mesh Generation · Digital Image Processing Techniques · Topological and Geometric Data Analysis
