Reaching 3-connectivity via edge-edge additions

Abstract

Given a graph G and a pair 〈e' e''〉 of distinct edges of G, an edge-edge addition on 〈e' e''〉 is an operation that turns G into a new graph G' by subdividing edges e' and e'' with a dummy vertex v' and v'', respectively, and by adding the edge (v',v''). In this paper, we show that any 2-connected simple planar graph with minimum degree δ(G) ≥ 3 and maximum degree Δ(G) can be augmented by means of edge-edge additions to a 3-connected planar graph G' with Δ(G') = Δ(G), where each edge of G participates in at most one edge-edge addition. This result is based on decomposing the input graph into its 3-connected components via SPQR-trees and on showing the existence of a planar embedding in which edge pairs from a special set share a common face. Our proof is constructive and yields a linear-time algorithm to compute the augmented graph. As a relevant application, we show how to exploit this augmentation technique to extend some classical NP-hardness results for bounded-degree 2-connected planar graphs to bounded-degree 3-connected planar graphs.