Augmenting Geometric Graphs with Matchings

We study noncrossing geometric graphs and their disjoint compatible geometric matchings. Given a cycle (a polygon) P we want to draw a set of pairwise disjoint straight-line edges with endpoints on the vertices of P such that these new edges neither cross nor contain any edge of the polygon. We prove NP-completeness of deciding whether there is such a perfect matching. For any n-vertex polygon, with n > 3, we show that such a matching with less than n/7 edges is not maximal, that is, it can be extended by another compatible matching edge. We also construct polygons with maximal compatible matchings with n/7 edges, demonstrating the tightness of this bound. Tight bounds on the size of a minimal maximal compatible matching are also obtained for the families of d-regular geometric graphs for each d in {0,1,2}. Finally we consider a related problem. We prove that it is NP-complete to decide whether a noncrossing geometric graph G admits a set of compatible noncrossing edges such that G together with these edges has minimum degree five.

[1]  Csaba D. Tóth,et al.  Disjoint compatible geometric matchings , 2011, SoCG '11.

[2]  David Rappaport,et al.  Computing simple circuits from a set of line segments , 1987, SCG '87.

[3]  Oswin Aichholzer,et al.  Compatible matchings in geometric graphs , 2011 .

[4]  Mark de Berg,et al.  Optimal Binary Space Partitions in the Plane , 2010, COCOON.

[5]  Alexander Pilz Augmentability to Cubic Graphs , 2012 .

[6]  Sang Won Bae,et al.  Shortcuts for the Circle , 2017, ISAAC.

[7]  Sergey Bereg,et al.  Compatible geometric matchings , 2007, Comput. Geom..

[8]  Csaba D. Tóth,et al.  Plane Geometric Graph Augmentation: A Generic Perspective , 2013 .

[9]  Klaus Jansen One Strike Against the Min-max Degree Triangulation Problem , 1992, Comput. Geom..

[10]  Michiel H. M. Smid,et al.  Minimizing the Continuous Diameter When Augmenting a Tree with a Shortcut , 2017, Workshop on Algorithms and Data Structures.

[11]  Ignaz Rutter,et al.  Regular Augmentation of Planar Graphs , 2014, Algorithmica.

[12]  Dimitrios M. Thilikos,et al.  A polynomial-time algorithm for Outerplanar Diameter Improvement , 2017, J. Comput. Syst. Sci..

[13]  Ioannis G. Tollis,et al.  Planar grid embedding in linear time , 1989 .

[14]  Günter Rote,et al.  Minimum-weight triangulation is NP-hard , 2006, JACM.

[15]  Csaba D. Tóth,et al.  Circumscribing Polygons and Polygonizations for Disjoint Line Segments , 2019, Discrete & Computational Geometry.

[16]  Godfried T. Toussaint,et al.  On computing simple circuits on a set of line segments , 1986, SCG '86.