Abstract
A combinatorial embedding \({\it \Pi}\) of a planar graph G = (V,E) is defined by the cyclic order of incident edges around each vertex in a planar drawing of G. The planar biconnectivity augmentation problem with fixed embedding (PBA-Fix) asks for a minimum edge set E′ ⊆ V×V that augments \({\it \Pi}\) to a combinatorial embedding \({\it \Pi}'\) of G + E′ such that G + E′ is biconnected and \({\it \Pi}\) is preserved, i.e., \({\it \Pi}'\) restricted to G yields again \({\it \Pi}\).
In this paper, we show that PBA-Fix is NP-hard in general, i.e., for not necessarily connected graphs, by giving a reduction from 3-PARTITION. For connected graphs, we present an \(\mathcal{O}(|V|(1+\alpha(|V|)))\) time algorithm solving PBA-Fix optimally. Moreover, we show that—considering each face of \({\it \Pi}\) separately—this algorithm meets the lower bound for the general biconnectivity augmentation problem proven by Eswaran and Tarjan [1].
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Gutwenger, C., Mutzel, P., Zey, B. (2009). Planar Biconnectivity Augmentation with Fixed Embedding. In: Fiala, J., Kratochvíl, J., Miller, M. (eds) Combinatorial Algorithms. IWOCA 2009. Lecture Notes in Computer Science, vol 5874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10217-2_29
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DOI: https://doi.org/10.1007/978-3-642-10217-2_29
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