Edge-Splitting and Edge-Connectivity Augmentation in Planar Graphs

Abstract

Let G = (V,E) be a k-edge-connected multigraph with a designated vertex s ∈ V which has even degree. A splitting operation at s replaces two edges (s, u) and (s, v) incident to s with a single edge (u, v). A set of splitting operations at s is called complete if there is no edge incident to s in the resulting graph. It is known by Lovász (1979) that there always exists a complete splitting at s such that the resulting graph G′ (neglecting the isolated vertex s) remains k-edge-connected. In this paper, we prove that, in the case where G is planar and k is an even integer or k = 3, there exists a complete splitting at s such that the resulting graph G′ remains k-edge-connected and planar, and present an O(|V|³ log|V|) time algorithm for finding such a splitting. However, for every odd k ≥ 5, there is a planar graph G with a vertex s which has no complete splitting at s which preserves both k-edge-connectivity and planarity. As an application of this result, we show that the problem of augmenting the edge-connectivity of a given outerplanar graph to an even integer k or to k = 3 can be solved in polynomial time.