Fixed-parameter Tractable Distances to Sparse Graph Classes

TL;DR

This paper proves that for various sparse graph classes and distance measures, the problem of computing the distance from a given graph is fixed-parameter tractable, using two general techniques related to nowhere dense classes and elimination distance.

Contribution

It introduces new fixed-parameter tractability results for measuring graph distance to sparse classes, based on slicewise definability and elimination distance techniques.

Findings

Distance to sparse classes is fixed-parameter tractable.

Uses slicewise nowhere dense and first-order definability concepts.

Establishes FPT for elimination distance to minor-closed classes.

Abstract

We show that for various classes C of sparse graphs, and several measures of distance to such classes (such as edit distance and elimination distance), the problem of determining the distance of a given graph G to C is fixed-parameter tractable. The results are based on two general techniques. The first of these, building on recent work of Grohe et al. establishes that any class of graphs that is slicewise nowhere dense and slicewise first-order definable is FPT. The second shows that determining the elimination distance of a graph G to a minor-closed class C is FPT.