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An $X$-TAR (token addition/removal) reconfiguration graph has as its verticessets that satisfy some property $X$, with an edge between two sets if one isobtained from the other by adding or removing one element. This paper considersthe $X$-TAR graph for $X-$ sets of vertices of a base graph $G$ where the$X$-sets of $G$ must satisfy certain conditions. Dominating sets, powerdominating sets, zero forcing sets, and positive semidefinite zero forcing setsare all examples of $X$-sets. For graphs $G$ and $G'$ with no isolatedvertices, it is shown that $G$ and $G'$ have isomorphic $X$-TAR reconfigurationgraphs if and only if there is a relabeling of the vertices of $G'$ such that$G$ and $G'$ have exactly the same $X$-sets. The concept of an $X$-irrelevantvertex is introduced to facilitate analysis of $X$-TAR graph isomorphisms.Furthermore, results related to the connectedness of the zero forcing TAR graphare given. We present families of graphs that exceed known lower bounds forconnectedness parameters.

Isomorphisms and properties of TAR reconfiguration graphs for zero forcing and other $X$-set parameters