How to use planarity efficiently: new tree-decomposition based algorithms

Authors:
Frederic Dorn
Affiliations:
Department of Informatics, University of Bergen, Bergen, Norway
Venue:
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
Year:
2007

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Abstract

We prove new structural properties for tree-decompositions of planar graphs that we use to improve upon the runtime of tree-decomposition based dynamic programming approaches for several NP-hard planar graph problems. We give for example the fastest algorithm for PLANAR DOMINATING SET of runtime 3tw ċ nO(1), when we take the treewidth tw as the measure for the exponential worst case behavior. We also introduce a tree-decomposition based approach to solve non-local problems efficiently, such as PLANAR HAMILTONIAN CYCLE in runtime 6tw ċ nO(1). From any input tree-decomposition, we compute in time O(nm) a tree-decomposition with geometric properties, which decomposes the plane into disks, and where the graph separators form Jordan curves in the plane.