Edge-Connectivity Augmentation of Simple Graphs

We consider the following variant of the edge-augmentation problem: Given a \(k\)-edge-connected graph with no loops or multiple edges, find a smallest edge set in the complement whose addition to \(G\) results in a \((k+1)\)-edge-connected graph. We establish the following dichotomy for this problem: If the complement of \(G\) contains a matching covering all vertices of \(G\)-degree \(k\) (and possibly more), then the complement also contains a matching whose addition to \(G\) results in a \((k+1)\)-edge-connected graph. A smallest matching which augments the minimum degree can be found, in polynomial time, by Edmonds’ matching algorithm, but it need not augment the edge-connectivity. Indeed, it is NP-hard to find a smallest edge-connectivity augmenting edge set, by a result of Tibor Jordán. On the other hand, if the complement of \(G\) contains no matching covering all vertices of \(G\)-degree \(k\), then the complement has a minimum degree augmenting path system consisting of paths of length 1 or 2. Again we can find such a path system with as few edges as possible by Edmonds’ matching algorithm. We can, in polynomial time, modify it to an edge-connectivity augmenting path system of paths of length 1 or 2 with the same number of edges, and this time it yields a smallest edge-connectivity augmenting set of edges. Combining these results, we conclude that a smallest edge-connectivity augmenting edge set in the complement of a \(k\)-regular, \(k\)-edge-connected simple graph has size \(n-m(\overline{G)}\), where \(n\) is the number of vertices of \(G\), and \(m(\overline{G)}\) is the size of a maximum matching in the complement of \(G\). Another corollary is that the complement of every simple noncomplete graph \(G\) with \(n\) vertices has a set of at most \(2n/3\) edges whose addition to \(G\) results in a graph of larger edge-connectivity, with equality holding if and only the complement of \(G\) is a disjoint union of 3-cycles.