On the Hardness and Approximability of Planar Biconnectivity Augmentation

Abstract

Given a planar graph G = (V,E), the planar biconnectivity augmentation problem (PBA) asks for an edge set E′ ⊆ V×V such that G + E′ is planar and biconnected. This problem is known to be $\mathcal{NP}$-hard in general; see [1]. We show that PBA is already $\mathcal{NP}$-hard if all cutvertices of G belong to a common biconnected component B ^*, and even remains $\mathcal{NP}$-hard if the SPQR-tree of B ^* (excluding Q-nodes) has a diameter of at most two. For the latter case, we present a new 5/3-approximation algorithm with runtime ${\mathcal{O}}(|V|^{2.5})$.

Though a 5/3-approximation of PBA has already been presented [12], we give a family of counter-examples showing that this algorithm cannot achieve an approximation ratio better than 2, thus the best known approximation ratio for PBA is 2.