Perfect 2-Distance Zero Forcing in Graphs

Let G be a graph. Then a 2-distance color change rule is defined as follows: If a vertex x ∈ V (G) is colored and has exactly one hop neighbor y is uncolored, then y will become colored. Moreover, let u, v, w ∈ V (G). If u 2-forces v and v 2-forces w, then we say that v and w are perfectly 2-forced by u, and this process can extend to a chain of 2-forcing initiated by a single vertex. In addition, a subset S of a vertex-set V (G) of G is called a perfect 2-distance zero forcing set of G if there exists s ∈ S such that s perfectly 2-forces all other vertices outside S. The minimum cardinality of a perfect 2-distance zero forcing set of G, denoted by Z2p(G), is called the perfect 2-distance zero forcing number of G. In this paper, this new parameter is introduced and initially investigated on some classes of graphs and on the join of two graphs. A particular variant of zero forcing called perfect co-zero forcing is defined to study the behavior of the perfect 2-distance zero forcing sets in the join of graphs. Characterizations of perfect 2-distance zero forcing sets are formulated and subsequently used to obtain some formulas for solving the perfect 2-distance zero forcing numbers of the join of some graphs.