A polynomial-time algorithm for Outerplanar Diameter Improvement

Cohen, Nathann; Kim, Eun Jung; Paul, Christophe; Sau, Ignasi; Thilikos, Dimitrios M.; Weller, Mathias

doi:10.1016/j.jcss.2017.05.016

Cited by 3 publications

(1 citation statement)

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“…Specifically it is NP-hard to decide whether a geometric graph can be augmented to a cubic geometric graph [13] and also whether an abstract planar graph can be augmented to a cubic planar graph (not preserving any fixed embedding) [8]. Besides the problems mentioned in that survey, decreasing the diameter [6] and the continuous setting (where every point along the edges of an embedded graph is considered as a vertex) received considerable attention [4,7].…”

Section: Introductionmentioning

confidence: 99%

Augmenting Geometric Graphs with Matchings

Pilz¹,

Rollin²,

Schlipf³

et al. 2020

Preprint

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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 so 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 ≥ 4, we show that such a matching with < 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 ∈ {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.

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