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.
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Zeh, N. (2004). Connectivity of Graphs Under Edge Flips. In: Hagerup, T., Katajainen, J. (eds) Algorithm Theory - SWAT 2004. SWAT 2004. Lecture Notes in Computer Science, vol 3111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27810-8_15
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DOI: https://doi.org/10.1007/978-3-540-27810-8_15
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