In this work, we study the zero forcing problem and some of its variants regarding the connectivity of the subgraph generated by the zero forcing set. A set Z of vertices from a graph G is said to be a zero forcing set of G if iteratively adding to it unique neighboring vertices of those vertices V(G)∖Z already in Z results in the entire vertex set V(G) of G. The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set of G. In this paper, we establish tight combinatorial bounds for zero forcing, total zero forcing and connected zero forcing for maximal outerplanar graphs. We also present a lower bound for zero forcing for near-triangulations.

The concepts of monitoring the elements of triangulation graphs by edges and faces at a distance k were defined by Hernández and Martins (2014, Electr. Notes Discrete Math., 46, 145-152). Furthermore, for any n-vertex maximal outerplanar graph, they provide combinatorial bounds when k = 2. In this paper, we continue their study, generalizing its results to any value of k. We prove that on maximal outerplanar graphs.