Loop zero forcing and Grundy domination in planar graphs and claw-free cubic graphs (Q6564843)

Zero forcing is a coloring process of the vertices of a simple, finite graph \(G\) according to the following rule. Pick a set \(S\subseteq V(G)\) and color each vertex of \(S\) blue and each vertex of \(V(G)-S\) white. The color change rule is that for each \(w\in V(G)-S\), where \(w\) is the only white neighbor of a blue vertex \(v\), then color \(w\) blue. If after applying the color change rule as many times as possible each vertex of \(G\) is blue, then we call \(S\) a zero forcing set of \(G\). The zero forcing number of \(G\), denoted by \(Z(G)\), is the minimum cardinality among all zero forcing sets of \(G\). The Grundy-like domination number is the dual invariant to the zero forcing number of a graph.\N\NA loop zero forcing set \(S\subseteq V(G)\) is also defined if the following is true. Initially, color all vertices of \(S\) blue and all vertices of \(V(G)-S\) white. Iteratively apply the loop zero forcing color change rule: if \(w\in V(G)\) is either 1) the only white neighbor of blue vertex \(v\), or 2) every neighbor of \(w\) is blue, then recolor \(w\) blue. If after applying the loop zero forcing color change rule as many times as possible all vertices of \(G\) are blue, then call \(S\) a loop zero forcing set of \(G\). \(Z_{\dot{\ell}}(G)\) denotes the minimum cardinality among all loop zero forcing sets of \(G\). The Grundy domination number of a graph is the dual invariant of loop zero forcing number of a graph.\N\NThe authors are motivated by the results of \textit{R. Davila} and \textit{M. A. Henning} [Bull. Malays. Math. Sci. Soc. (2) 43, No. 1, 673--688 (2020; Zbl 1433.05117)] and consider the loop zero forcing number of 2-connected, claw-free cubic graphs. A similar technique of \textit{S. E. Anderson} and \textit{K. Kuenzel} [Discrete Math. 345, No. 11, Article ID 113113, 10 p. (2022; Zbl 1495.05206)] is used. The following upper bound for \(Z(G)\) in the case where \(G\) is a 2-edge-connected, claw-free cubic graph is established:\N\NIf \(G\) is not a ring of diamonds, then\N\[\NZ_{\dot{\ell}}(G)\leq Z(G)\leq \left\lceil\frac{5|V(G)|}{18}\right\rceil+1.\N\]\NIf \(G\) is a ring of diamonds, then\N\[\NZ_{\dot{\ell}}(G)=Z(G)= \frac{|V(G)|}{4}+2.\N\]\NThe authors also consider the loop zero forcing number of certain types of planar graphs. They derive a lower bound for the zero forcing number of maximal outerplanar graphs as well as show that in certain cases the loop zero forcing number and zero forcing number of maximal outerplanar graphs coincide.