Reduction algorithms for constructing solutions in graphs with small treewidth

Abstract

This paper presents some new ideas and results on graph reduction applied to graphs with bounded treewidth. Arnborg et al. [2] have shown that many decision problems on graphs can be solved in linear time on graphs with bounded treewidth, by using a finite set of reduction rules. Bodlaender [5] has shown that a number of optimization problems can also be solved in this way. We show that these methods can be extended to solve the construction variants of many of these problems on graphs with bounded treewidth. For example, the construction variants of decision problems that are definable in monadic second order logic can be solved in this way, and construction variants of Independent set and Hamiltonian completion number can be solved in this way.

We also show that the results of [7] can be applied to our reduction algorithms, which results in parallel 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 (where n is the number of vertices of the graph).

This research was partially supported by the Foundation for Computer Science (S.I.O.N) of the Netherlands Organization for Scientific Research (N.W.O.) and partially by the ESPRIT Basic Research Actions of the EC under contract 7141 (project ALCOM II). A full version of this paper