Connectivity of Graphs Under Edge Flips

We study the following problem: Given a set V of n vertices and a set E of m edge pairs, we define a graph family G(V, E) as the set of graphs that have vertex set V and contain exactly one edge from every pair in E . We want to find a graph in G(V, E) that has the minimal number of connected components. We show that, if the edge pairs in E are non-disjoint, the problem is NP-hard. This is true even if an edge is not allowed to appear in more than two edge pairs and the union of the graphs in G(V, E) is planar. if the edge pairs are disjoint, we provide an O(nm)-time algorithm that finds a graph in G(V, E) with the minimal number of connected components. Our proof of the latter statement is obtained by flipping edges in the graphs in G(V, E), where the flip of an edge e in a graph G ∈ G(V, E) removes e from G and inserts the other edge in the pair in E that contains e. We explore also the question whether any graph in G(V, E) can be transformed into any other graph in G(V, E) using edge flips while guaranteeing an upper bound on the number of connected components of every intermediate graph.