TAR reconfiguration for vertex set parameters

TL;DR

This paper surveys and extends results on token addition and removal reconfiguration graphs for various vertex set parameters, including domination, zero forcing, and independence, highlighting new applications to maximizing parameters and graph connectedness.

Contribution

It broadens the scope of TAR reconfiguration results to include $Y$-sets and maximization parameters, and explores the necessary axioms for these results.

Findings

Main results apply to skew zero forcing and vertex covering.

TAR reconfiguration graphs for these parameters are characterized.

Connectedness in reconfiguration graphs extends to $X$-set and $Y$-set parameters.

Abstract

This paper surveys results about token addition and removal (TAR) reconfiguration for several well-known vertex set parameters including domination, power domination, standard zero forcing, and PSD zero forcing. We also expand the range of parameters to which universal $X$-set TAR graph results apply, for $X$-sets and their TAR graphs introduced in [B. Bjorkman, C. Bozeman, D. Ferrero, M. Flagg, C. Grood, L. Hogben, B. Jacob, C. Reinhart, Power domination reconfiguration, arXiv:2201.01798] and [N.H. Bong, J. Carlson, B. Curtis, R. Haas, L. Hogben, Isomorphisms and properties of TAR reconfiguration graphs for zero forcing and other $X$-set parameters, \emph{Graphs Combin.} {39} (2023), Paper No. 86]. Here we examine which of the $X$-set axioms are needed for which results. The main results apply to skew zero forcing and vertex covering, and results about TAR reconfiguration graphs of these parameters are presented. While $X$-sets are defined for parameters that take the minimum cardinality over the $X$-sets of a graph, and $X$-set results are restricted to such minimizing parameters, our expansion of the universal perspective allows these results to be applied to parameters that take the maximum value among relevant sets, called $Y$-sets. Maximizing parameters to which the main results apply include independence number, (upper) irredundance number, and (upper) zero forcing irredundance number, and failed zero forcing number; TAR reconfiguration results are presented for these parameters. We also show that the equivalence of connectedness in certain token jumping reconfiguration graphs and certain TAR reconfiguration graphs for independent sets established in [M. Kami\'nski, P. Medvedev, M. Milani\v c. Complexity of independent set reconfiguration problems. {\em J. Theoretical Computer Science} 439 (2012), 9--15.] extends to $X$-set and $Y$-set parameters.