Recovering sparse graphs

TL;DR

This paper introduces a fixed parameter algorithm for approximating original graphs from modified d-degenerate graphs, with applications to fixed parameter tractability in first order model checking on sparse graph classes.

Contribution

It presents a novel fixed parameter algorithm for recovering graphs from certain modifications, advancing understanding of model checking in sparse graph classes.

Findings

Algorithm recovers original graphs with bounded vertex differences.

Fixed parameter tractability established for first order model checking.

Applicable to classes of graphs derived from bounded expansion graphs.

Abstract

We construct a fixed parameter algorithm parameterized by d and k that takes as an input a graph G' obtained from a d-degenerate graph G by complementing on at most k arbitrary subsets of the vertex set of G and outputs a graph H such that G and H agree on all but f(d,k) vertices. Our work is motivated by the first order model checking in graph classes that are first order interpretable in classes of sparse graphs. We derive as a corollary that if G_0 is a graph class with bounded expansion, then the first order model checking is fixed parameter tractable in the class of all graphs that can obtained from a graph G from G_0 by complementing on at most k arbitrary subsets of the vertex set of G; this implies an earlier result that the first order model checking is fixed parameter tractable in graph classes interpretable in classes of graphs with bounded maximum degree.