Loop Zero Forcing and Grundy Domination in Planar Graphs and Claw-Free Cubic Graphs

Abstract

Given a simple, finite graph with vertex set V(G), we define a zero forcing set of G as follows. Choose \(S\subseteq V(G)\) and color all vertices of S blue and all vertices in \(V(G) - S\) white. The color change rule is if w is the only white neighbor of blue vertex v, then we change the color of w from white to blue. If after applying the color change rule as many times as possible eventually every vertex of G is blue, we call S a zero forcing set of G. Z(G) denotes the minimum cardinality of a zero forcing set. We show that if G is 2-edge-connected, claw-free, and cubic, then . We also study a similar graph invariant known as the loop zero forcing number of a graph G which happens to be the dual invariant to the Grundy domination number of G. Specifically, we study the loop zero forcing number in two particular types of planar graphs.