How to Use Planarity Efficiently: New Tree-Decomposition Based Algorithms

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 3 · n, 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 6 ·n. 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.