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3-path-step graph operator; tree; unicyclic graph
E. Prisner in his book Graph Dynamics defines the $k$-path-step operator on the class of finite graphs. The $k$-path-step operator (for a positive integer $k$) is the operator $S^{\prime }_k$ which to every finite graph $G$ assigns the graph $S^{\prime }_k(G)$ which has the same vertex set as $G$ and in which two vertices are adjacent if and only if there exists a path of length $k$ in $G$ connecting them. In the paper the trees and the unicyclic graphs fixed in the operator $S^{\prime }_3$ are studied.
[1] F. Escalante, L. Montejano: Trees and $n$-path invariant graphs, Abstract. Graph Theory Newsletter 33 (1974).
[2] E. Prisner: Graph Dynamics. Longman House, Burnt Mill, Harlow, 1998. MR 1379114
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