Minimizing the Stabbing Number of Matchings, Trees, and Triangulations

Computing Triangulations with Minimum Stabbing Number

For a given point set P or a polygon P , we consider the problem of finding a triangulation T with minimum stabbing number, i.e., a triangulation such that the maximal number of segments hit by any ray going through T is minimized. We prove that this problem is NP-hard; this differs from the problem of triangulating a polygon with minimum edge weight, which is solvable in polynomial time with a...

On R-trees with Low Stabbing Number

ar X iv : c s / 03 10 03 4 v 2 [ cs . C G ] 7 S ep 2 00 5 Minimizing the Stabbing Number of Matchings , Trees , and Triangulations ∗

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the or...

The number of maximal matchings in polyphenylene chains

A matching is maximal if no other matching contains it as a proper subset. Maximal matchings model phenomena across many disciplines, including applications within chemistry. In this paper, we study maximal matchings in an important class of chemical compounds: polyphenylenes. In particular, we determine the extremal polyphenylene chains in regards to the number of maximal matchings. We also de...

Graphs of Triangulations and Perfect Matchings

Given a set P of points in the plane, the graph of triangulations T (P ) of P has a vertex for every triangulation of P , and two of them are adjacent if they differ by a single edge exchange. We prove that the subgraph TM(P ) of T (P ), consisting of all triangulations of P that admit a perfect matching, is connected. A main tool in our proof is a result of independent interest, namely that th...