On Reconfigurability of Target Sets

TL;DR

This paper investigates the computational complexity of reconfiguring target sets in graphs, proving PSPACE-completeness in many cases and providing polynomial algorithms for specific graph classes like trees and degree-2 graphs.

Contribution

It establishes the PSPACE-completeness of Target Set Reconfiguration in various graph classes and offers efficient algorithms for degree-2 graphs and trees, extending previous results.

Findings

PSPACE-completeness on bipartite planar graphs of degree 3 and 4

Polynomial-time algorithm for graphs of maximum degree 2

Polynomial-time algorithm for trees

Abstract

We study the problem of deciding reconfigurability of target sets of a graph. Given a graph $G$ with vertex thresholds $\tau$, consider a dynamic process in which vertex $v$ becomes activated once at least $\tau(v)$ of its neighbors are activated. A vertex set $S$ is called a target set if all vertices of $G$ would be activated when initially activating vertices of $S$. In the Target Set Reconfiguration problem, given two target sets $X$ and $Y$ of the same size, we are required to determine whether $X$ can be transformed into $Y$ by repeatedly swapping one vertex in the current set with another vertex not in the current set preserving every intermediate set as a target set. In this paper, we investigate the complexity of Target Set Reconfiguration in restricted cases. On the hardness side, we prove that Target Set Reconfiguration is PSPACE-complete on bipartite planar graphs of degree $3$ and $4$ and of threshold $2$, bipartite $3$-regular graphs and planar $3$-regular graphs of threshold $1$ and $2$, and split graphs, which is in contrast to the fact that a special case called Vertex Cover Reconfiguration is in P for the last graph class. On the positive side, we present a polynomial-time algorithm for Target Set Reconfiguration on graphs of maximum degree $2$ and trees. The latter result can be thought of as a generalization of that for Vertex Cover Reconfiguration.