Positive-Influence via Roman Domination in Graphs

In this paper, we introduce and study the positive-influence Roman dominating function (PIRDF), a novel graph-theoretic concept that unifies Roman domination with positive-influence domination. A labeling f : V(G) → {0,1,2} is a PIRDF if it satisfies the Roman domination condition and the set of positively labeled vertices forms a positive-influence dominating set, meaning every unlabeled vertex has at least half of its neighbors positively labeled. The positive-influence Roman domination number, \( \gamma_R^{PI}(G) \), is defined as the minimum weight of a PIRDF of G. We establish several tight bounds. We prove that the associated decision problem is NP-complete for general graphs, and show that it is solvable in linear time for graph classes of bounded clique-width via a LinEMSOL₁ formulation. Exact values of \( \gamma_R^{PI}(G) \) are computed for several standard graph families, including paths, cycles, complete graphs, complete bipartite graphs, star graphs, and friendship graphs. In addition, degree-based bounds are derived, and graphs attaining the lower bound \( \gamma_R^{PI}(G) = 1 + \lceil \delta/2 \rceil \) are fully characterized.