Abstract
Given a set V of n vertices and a set \(\mathcal{E}\) of m edge pairs, we define a graph family \(\mathcal{G}(V, \mathcal{E})\) as the set of graphs that have vertex set V and contain exactly one edge from every pair in \(\mathcal{E}\). We want to find a graph in \(\mathcal{G}(V, \mathcal{E})\) that has the minimal number of connected components. We show that, if the edge pairs in \(\mathcal{E}\) are non-disjoint, the problem is NP-hard even if the union of the graphs in \(\mathcal{G}(V, \mathcal{E})\) is planar. If the edge pairs are disjoint, we provide an \(\mathcal{O}(n^2 m)\)-time algorithm that finds a graph in \(\mathcal{G}(V, \mathcal{E})\) with the minimal number of connected components.KeywordsSpan TreeDelaunay TriangulationConjunctive Normal FormLocal TransformationCommon CycleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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