Mutual witness proximity graphs

• Mutual witness proximity graphs on point sets A and B   are defined. Set n=|A|+|B|n=|A|+|B|.
• Two mutual witness Delaunay graphs together contain ≥⌈(n−2)/2⌉≥⌈(n−2)/2⌉ edges; this is tight.
• If two mutual witness Delaunay graphs are complete, A and B are circularly separable.
• If two mutual witness Gabriel graphs are complete, A and B are linearly separable.
• Two mutual witness rectangle graphs may be complete for linearly inseparable A and B.

This paper describes one variation on witness proximity graphs called mutual witness proximity graphs. Two witness proximity graphs are said to be mutual when, given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B   is the witness set of the first graph and the vertex set of the second one. We show that in the union of two mutual witness Delaunay graphs, there are always at least ⌈n−22⌉ edges, where n=|A|+|B|n=|A|+|B|, which is tight in the worst case. We also show that if two mutual witness Delaunay graphs are complete, then the sets A and B are circularly separable; if two mutual witness Gabriel graphs are complete, then the sets A and B are linearly separable; but two mutual witness rectangle graphs might be complete, with A and B not linearly separable.