[PDF] Descriptive Complexity, Canonisation, and Definable Graph Structure Theory | Semantic Scholar

This paper proves two general theorems lifting definability results from the pieces of a tree decomposition of a graph to the whole graph and can be used to prove that the class of all graphs without a K5-minor is definable infixed- point logic and that fixed-point logic with counting captures polynomial time on this class.

It is proved that graphs with excluded minors can be decomposed into pieces arranged in a treelike structure, together with a linear order of each of the pieces, and the decomposition and the linear orders on the pieces are definable in fixed-point logic (without counting).

The analysis of IFP+C-formulas will help to clarify the expressive power of I FP+C; in particular, it is hoped that a normal form is derived under first-order reductions, which may give a better understanding of polynomial time.

It is proved that for every class C of graphs with excluded minors there is a k such that a simple combinatorial algorithm, namely "the k-dimensional Weisfeiler--Lehman algorithm," decides isomorphism of graphs in C in polynomial time.

This work defines a canonical form of interval graphs using a type of modular decomposition, which is different from the method of tree decomposition that is used in most known capturing results for other graph classes, specifically those defined by forbidden minors.

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