Unsolved problems in visibility graphs of points, segments, and polygons
ACM Computing Surveys (CSUR)
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The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and vertex-connectivity of visibility graphs. Unless all its vertices are collinear, a visibility graph has diameter at most 2, and so it follows by a result of Plesník (Acta Fac. Rerum Nat. Univ. Comen. Math. 30:71–93, 1975) that its edge-connectivity equals its minimum degree. We strengthen the result of Plesník by showing that for any two vertices v and w in a graph of diameter 2, if deg(v)≤deg(w) then there exist deg(v) edge-disjoint vw-paths of length at most 4. For vertex-connectivity, we prove that every visibility graph with n vertices and at most ℓ collinear vertices has connectivity at least $\frac{n-1}{\ell-1}$, which is tight. We also prove the qualitatively stronger result that the vertex-connectivity is at least half the minimum degree. Finally, in the case that ℓ=4 we improve this bound to two thirds of the minimum degree.