Forcing and domination in graphs

Let $G$ be a connected graph. We investigated structural relationships between the zero forcing number $Z(G)$ and connected domination. Our main result proved that the complement of every minimal connected dominating set formed a zero forcing set, establishing the converse direction to a known result of Amos et al. and yielding the inequality $Z(G)+{\gamma }_c(G)\le |V(G)|$. We further studied graphs attaining equality and showed that several natural graph families, including complete multipartite graphs, satisfied it. Motivated by the duality between forcing and domination, we introduced the dom-forcing number $F_d(G)$, the minimum size of a set that was both dominating and zero forcing. We established sharp general bounds, determined its exact values for paths, stars, complete graphs, and complete bipartite graphs, and characterized when it equaled the zero forcing number or the domination number. Finally, we discussed several open problems related to forcing-domination dualities.