Problems Easy for Tree-Decomposable Graphs (Extended Abstract)

Using a variation of the interpretability concept we show that all graph properties definable in monadic second order logic (MS properties) with quantification over vertex and edge sets can be decided in linear time for classes of graphs of fixed bounded tree-width, giving an alternative proof of a recent result by Courcelle. We allow graphs with directed and/or undirected edges, labeled on edges and/or vertices with labels taken from a finite set. We extend MS properties to Extended Monadic Second-order (EMS) problems involving counting or summing evaluations given with the graph over sets definable in monadic second order logic. Our tecnique allowes us to solve also some EMS problems in linear time or in polynomial or pseudopolynomial time for classes of graphs of fixed bounded tree-width. Most problems for wich linear time algorithms for graphs of bounded tree width where previously known to cxist, and many others, are EMS problems. Research supported by the Swedish Natural Sciences Research Council and the Swedish Board for Technological Development. t Department of Numerical Analysis and Computing Science, The Royal Institute of Technology, S-100 44 Stockholm, Sweden. Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut fgr Mathematik, Mohrenstrasse 39, DDR Berlin 1086.