Reconfiguration on nowhere dense graph classes

TL;DR

This paper investigates the computational complexity of reconfiguration problems for distance-based independent and dominating sets on graph classes, establishing polynomial kernels for nowhere dense classes and hardness results for somewhere dense classes.

Contribution

It proves the existence of polynomial kernels for reconfiguration problems on nowhere dense graph classes and shows W[1]-hardness on somewhere dense classes, delineating the boundary of tractability.

Findings

Polynomial kernels exist for reconfiguration problems on nowhere dense classes.

Reconfiguration problems are W[1]-hard on somewhere dense classes.

The boundary between tractability and hardness aligns with nowhere and somewhere denseness.

Abstract

Let $\mathcal{Q}$ be a vertex subset problem on graphs. In a reconfiguration variant of $\mathcal{Q}$ we are given a graph $G$ and two feasible solutions $S_s, S_t\subseteq V(G)$ of $\mathcal{Q}$ with $|S_s|=|S_t|=k$. The problem is to determine whether there exists a sequence $S_1,\ldots,S_n$ of feasible solutions, where $S_1=S_s$, $S_n=S_t$, $|S_i|\leq k\pm 1$, and each $S_{i+1}$ results from $S_i$, $1\leq i<n$, by the addition or removal of a single vertex. We prove that for every nowhere dense class of graphs and for every integer $r\geq 1$ there exists a polynomial $p_r$ such that the reconfiguration variants of the distance-$r$ independent set problem and the distance-$r$ dominating set problem admit kernels of size $p_r(k)$. If $k$ is equal to the size of a minimum distance-$r$ dominating set, then for any fixed $\epsilon>0$ we even obtain a kernel of almost linear size $\mathcal{O}(k^{1+\epsilon})$. We then prove that if a class $\mathcal{C}$ is somewhere dense and closed under taking subgraphs, then for some value of $r\geq 1$ the reconfiguration variants of the above problems on $\mathcal{C}$ are $\mathsf{W}[1]$-hard (and in particular we cannot expect the existence of kernelization algorithms). Hence our results show that the limit of tractability for the reconfiguration variants of the distance-$r$ independent set problem and distance-$r$ dominating set problem on subgraph closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.