Another Look of Rings Domination in Ladder Graph

Abstract

For a nontrivial connected graph \(G\) with no isolated vertex, a nonempty subset \(D \subseteq V(G)\) is a rings dominating set if each vertex \(v \in V-D\) is adjacent to at least two vertices in \(V-D\). Thus, the dominating set \(D\) of \(V(G)\) is a rings dominating set if for all \(v \in V-D,|N(v) \cap(V-D)| \geq 2\). The cardinality of minimum rings dominating set of \(G\) is the rings domination number of \(G\), denoted by \(\gamma_{r_i}\) whereas the cardinality of maximum rings dominating set is the upper rings domination number and is denoted by \(\gamma_{r i}^{\prime}\). Here, we determine how the rings dominating set is constructed in the ladder graph with the inclusion of generated conditions for this type of domination and give new approach for its parameter.