Abstract:

Let \( G = (V(G),E(G)) \) be a connected graph and let \( \alpha = (V_0,V_1,V_2) \) be a CvRDF on \( G \). A function \( \alpha \) is an outer-connected convex Roman dominating function (OCCvRDF) on \( G \) provided that for every \( v \in V_0 \), there exists \( u \in V_2 \) such that \( uv \in E(G) \), \( V_1 \cup V_2 \) is a convex set on \( G \), and \( \langle V_0 \rangle \) is a connected subgraph. The weight of OCCvRDF \( \alpha \) is denoted by \( \widetilde{\omega}_G^{cCvR}(\alpha) \) and is defined as \( \widetilde{\omega}_G^{cCvR}(\alpha) = \sum_{z \in V(G)} \alpha(z) = |V_1| + 2|V_2| \). The outer-connected convex Roman domination number of graph \( G \) is denoted by \( \widetilde{\gamma}_{cCvR}(G) \) and is defined as the minimum weight of an OCCvRDF \( \alpha \) on \( G \), which can be written as \( \widetilde{\gamma}_{cCvR}(G) = \min\{\widetilde{\omega}_G^{cCvR}(\alpha) : \alpha \text{ is an OCCvRDF on } G\} \). Each OCCvRDF \( \alpha \) on \( G \) satisfying the equation \( \widetilde{\omega}_G^{cCvR}(\alpha) = \widetilde{\gamma}_{cCvR}(G) \) is called a \( \widetilde{\gamma}_{cCvR} \)-function on \( G \). In this paper, we introduce a new study of convex Roman domination in graphs, called the outer-connected convex Roman dominating function, and give some of its graph-theoretic properties.