Connectivity of graphs under edge flips

Abstract

Given a set V of n vertices and a set ε of m edge pairs, we define a graph family G(V, ε) as the set of graphs that have vertex set V and contain exactly one edge from every pair in ε. We want to find a graph in G(V, ε) that has the minimal number of connected components. We show that, if the edge pairs in ε are non-disjoint, the problem is NP-hard even if the union of the graphs in G(V, ε) is planar. If the edge pairs are disjoint, we provide an script O sign(n2m)-time algorithm that finds a graph in G(V, ε) with the minimal number of connected components. © Springer-Verlag Berlin Heidelberg 2004.