Finding a smallest augmentation to biconnect a graph
SIAM Journal on Computing
SIAM Journal on Computing
A new approximation algorithm for the planar augmentation problem
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Planar Polyline Drawings with Good Angular Resolution
GD '98 Proceedings of the 6th International Symposium on Graph Drawing
A Linear Time Implementation of SPQR-Trees
GD '00 Proceedings of the 8th International Symposium on Graph Drawing
An O(v|v| c |E|) algoithm for finding maximum matching in general graphs
SFCS '80 Proceedings of the 21st Annual Symposium on Foundations of Computer Science
Connectivity augmentation in planar straight line graphs
European Journal of Combinatorics
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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.