Reduction Algorithms for Graphs of Small Treewidth

AbstractThis paper presents a number of new ideas and results on graph reduction applied to graphs of bounded treewidth. S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese (J. Assoc. Comput. Mach.40, 1134–1164 (1993)) have shown that many decision problems on graphs can be solved in linear time on graphs of bounded treewidth, using a finite set of reduction rules. These algorithms can be used to solve problems on graphs of bounded treewidth without the need to obtain a tree decomposition of the input graph first. We show that the reduction method can be extended to solve the construction variants of many decision problems on graphs of bounded treewidth, including all problems definable in monadic second order logic. We also show that a variant of these reduction algorithms can be used to solve (constructive) optimization problems in O(n) time. For example, optimization and construction variants of INDEPENDENT SET and HAMILTONIAN COMPLETION NUMBER can be solved in this way on graphs of small treewidth. Additionally, we show that the results of H. L. Bodlaender and T. Hagerup (SIAM J. Comput.27, 1725–1746 (1998)) can be applied to our reduction algorithms, which results in parallel reduction algorithms that use O(n) operations and O(log n log* n) time on an EREW PRAM, or O(log n) time on a CRCW PRAM