Bounded-Degree Polyhedronization of Point Sets

Abstract In 1994 Grünbaum showed that, given a point set S in R 3 , it is always possible to construct a polyhedron whose vertices are exactly S. Such a polyhedron is called a polyhedronization of S. Agarwal et al. extended this work in 2008 by showing that there always exists a polyhedronization that can be decomposed into a union of tetrahedra (tetrahedralizable). In the same work they introduced the notion of a serpentine polyhedronization for which the dual of its tetrahedralization is a chain. In this work we present a randomized algorithm running in O(n log 6 n) expected time that constructs a serpentine polyhedronization that has vertices with degree at most 7, answering an open question by Agarwal et al

A vertex-face assignment for plane graphs

For any planar straight line graph (Pslg), there is a vertexface assignment such that every vertex is assigned to at most two adjacent faces, and every face is assigned to all its reflex vertices and one more incident vertex. The existence of such an assignment implies, in turn, that any Pslg can be augmented to a connected Pslg such that the degree of every vertex increases by at most two

Abstract Disjoint Segments have Convex Partitions with 2-Edge Connected Dual Graphs

The empty space around n disjoint line segments in the plane can be partitioned into n + 1 convex faces by extending the segments in some order. The dual graph of such a partition is the plane graph whose vertices correspond to the n+1 convex faces, and every segment endpoint corresponds to an edge between the two incident faces on opposite sides of the segment. We construct, for every set of n disjoint line segments in the plane, a convex partition whose dual graph is 2-edge connected.