On computing graph minor obstruction sets

Abstract The Graph Minor Theorem of Robertson and Seymour establishes nonconstructively that many natural graph properties are characterized by a finite set of forbidden substructures, the obstructions for the property. We prove several general theorems regarding the computation of obstruction sets from other information about a family of graphs. The methods can be adapted to other partial orders on graphs, such as the immersion and topological orders. The algorithms are in some cases practical and have been implemented. Two new technical ideas are introduced. The first is a method of computing a stopping signal for search spaces of increasing pathwidth. This allows obstruction sets to be computed without the necessity of a prior bound on maximum obstruction width. The second idea is that of a second order congruence for a graph property. This is an equivalence relation defined on finite sets of graphs that generalizes the recognizability congruence that is defined on single graphs. It is shown that the obstructions for a graph ideal can be effectively computed from an oracle for the canonical second-order congruence for the ideal and a membership oracle for the ideal. It is shown that the obstruction set for a union F = F 1 ∪ F 2 of minor ideals can be computed from the obstruction sets for F 1 and F 2 if there is at least one tree that does not belong to the intersection of F 1 and F 2 . As a corollary, it is shown that the set of intertwines of an arbitrary graph and a tree are effectively computable.

[1]  Jens Lagergren.  An upper bound on the size of an obstruction , 1991, Graph Structure Theory.

[2]  P. Seymour,et al.  Surveys in combinatorics 1985: Graph minors – a survey , 1985 .

[3]  Robin Thomas,et al.  Quickly Excluding a Planar Graph , 1994, J. Comb. Theory, Ser. B.

[4]  Paul D. Seymour,et al.  Graph minors. I. Excluding a forest , 1983, J. Comb. Theory, Ser. B.

[5]  Detlef Seese,et al.  Monadic Second Order Logic, Tree Automata and Forbidden Minors , 1990, CSL.

[6]  Michael R. Fellows,et al.  Obstructions to Within a Few Vertices or Edges of Acyclic , 1995, WADS.

[7]  Michael R. Fellows,et al.  On search decision and the efficiency of polynomial-time algorithms , 1989, STOC '89.

[8]  Michael R. Fellows,et al.  A Simple Linear-Time Algorithm for Finding Path-Decompositions of Small Width , 1994, Inf. Process. Lett..

[9]  R. Downey,et al.  Parameterized Computational Feasibility , 1995 .

[10]  Bruno Courcelle,et al.  Equivalent definitions of recognizability for sets of graphs of bounded tree-width , 1996, Mathematical Structures in Computer Science.

[11]  C. Nash-Williams.  On well-quasi-ordering infinite trees , 1963, Mathematical Proceedings of the Cambridge Philosophical Society.

[12]  Michael R. Fellows,et al.  DNA Physical Mapping: Three Ways Difficult , 1993, ESA.

[13]  Russell Impagliazzo,et al.  Computing planar intertwines , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[14]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[15]  Michael J. Dinneen,et al.  A Characterization of Graphs with Vertex Cover up to Five , 1994, ORDAL.

[16]  Michael A. Langston,et al.  obstruction Set Isolation for the Gate Matrix Layout Problem , 1994, Discret. Appl. Math..

[17]  Michael R. Fellows,et al.  Nonconstructive tools for proving polynomial-time decidability , 1988, JACM.

[18]  Michael R. Fellows,et al.  Finite automata, bounded treewidth, and well-quasiordering , 1991, Graph Structure Theory.

[19]  Michael R. Fellows,et al.  Two Strikes Against Perfect Phylogeny , 1992, ICALP.

[20]  Michael R. Fellows,et al.  An analogue of the Myhill-Nerode theorem and its use in computing finite-basis characterizations , 1989, 30th Annual Symposium on Foundations of Computer Science.

[21]  S. Arnborg,et al.  Finding Minimal Forbidden Minors Using a Finite Congruence , 1991, ICALP.

[22]  Robin Thomas,et al.  Quickly excluding a forest , 1991, J. Comb. Theory, Ser. B.

[23]  Andrzej Proskurowski.  Graph reductions, and techniques for finding minimal forbidden minors , 1991, Graph Structure Theory.