Correlation Clustering and Two-Edge-Connected Augmentation for Planar Graphs

We study two problems. In correlation clustering, the input is a weighted graph, where every edge labelled either $$\langle +\rangle $$ or -\rangle according to whether its endpoints are in same category different categories. The goal produce partition of vertices into categories that tries respect labels edges. two-edge-connected augmentation, graph and subset R edges graph. minimum weight S such for R, $$R\cup S$$ . this paper, we these problems under restriction must be planar. give an approximation-preserving reduction from clustering on planar graphs augmentation graphs. polynomial-time approximation scheme (PTAS) latter problem, yielding PTAS former problem as well. employs brick decompositions, which have been used previous schemes graphs, but way it uses decompositions fundamentally uses.