On Plane Constrained Bounded-Degree Spanners

Abstract

Let P be a set of points in the plane and S a set of non-crossing line segments with endpoints in P. The visibility graph of P with respect to S, denoted $\mathord{\it Vis}(P,S)$, has vertex set P and an edge for each pair of vertices u,v in P for which no line segment of S properly intersects uv. We show that the constrained half-θ ₆-graph (which is identical to the constrained Delaunay graph whose empty visible region is an equilateral triangle) is a plane 2-spanner of $\mathord{\it Vis}(P,S)$. We then show how to construct a plane 6-spanner of $\mathord{\it Vis}(P,S)$ with maximum degree 6 + c, where c is the maximum number of segments adjacent to a vertex.

Research supported in part by NSERC and the Danish Council for Independent Research. Due to space constraints, some proofs are omitted and available in the full version of this paper.