Encompassing colored planar straight line graphs

Authors:
Ferran Hurtado;Mikio Kano;David Rappaport;Csaba D. Tóth
Affiliations:
Departament de Matèmatica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain;Department of Computer and Information Sciences, Ibaraki University, Hitachi, Japan;School of Computing, Queen's University, Kingston, ON, Canada;Department of Mathematics, MIT, Cambridge, MA, USA
Venue:
Computational Geometry: Theory and Applications
Year:
2008

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Abstract

Consider a planar straight line graph (PSLG), G, with k connected components, k=2. We show that if no component is a singleton, we can always find a vertex in one component that sees an entire edge in another component. This implies that when the vertices of G are colored, so that adjacent vertices have different colors, then (1) we can augment G with k-1 edges so that we get a color conforming connected PSLG; (2) if each component of G is 2-edge connected, then we can augment G with 2k-2 edges so that we get a 2-edge connected PSLG. Furthermore, we can determine a set of augmenting edges in O(nlogn) time. An important special case of this result is that any red-blue planar matching can be completed into a crossing-free red-blue spanning tree in O(nlogn) time.