
Circumscribing Polygons and Polygonizations for Disjoint Line Segments
Given a planar straightline graph G=(V,E) in R^2, a circumscribing poly...
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Plane augmentation of plane graphs to meet parity constraints
A plane topological graph G=(V,E) is a graph drawn in the plane whose ve...
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Minimum maximal matchings in cubic graphs
We prove that every connected cubic graph with n vertices has a maximal ...
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On Compatible Triangulations with a Minimum Number of Steiner Points
Two vertexlabelled polygons are compatible if they have the same clockw...
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(k,p)Planarity: A Generalization of Hybrid Planarity
A graph G is (k,p)planar if its vertices can be partitioned into cluste...
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Minimal Representations of Order Types by Geometric Graphs
In order to have a compact visualization of the order type of a given po...
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Complexity of Finding Perfect Bipartite Matchings Minimizing the Number of Intersecting Edges
Consider a problem where we are given a bipartite graph H with vertices ...
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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 straightline edges with endpoints on the vertices of P such that these new edges neither cross nor contain any edge of the polygon. We prove NPcompleteness of deciding whether there is such a perfect matching. For any nvertex 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 dregular geometric graphs for each d in 0,1,2. Finally we consider a related problem. We prove that it is NPcomplete 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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