Degree-constrained decompositions of graphs: bounded treewidth and planarity

Authors:
Cristina Bazgan;Zsolt Tuza;Daniel Vanderpooten
Affiliations:
LAMSADE, Université Paris-Dauphine, Paris Cedex, France;Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, Kende u., Hungary and Department of Computer Science, University of Veszprém, Hungary;LAMSADE, Université Paris-Dauphine, Paris Cedex, France
Venue:
Theoretical Computer Science
Year:
2006

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Abstract

We study the problem of decomposing the vertex set V of a graph into two nonempty parts V1, V2 which induce subgraphs where each vertex v ∈ V1 has degree at least a(v) inside V1 and each v ∈ V2 has degree at least b(v) inside V2. We give a polynomial-time algorithm for graphs with bounded treewidth which decides if a graph admits a decomposition, and gives such a decomposition if it exists. This result and its variants are then applied to designing polynomial-time approximation schemes for planar graphs where a decomposition does not necessarily exist but the local degree conditions should be met for as many vertices as possible.