Double power domination in graphs

Monitoring an electrical network using minimal measurement devices is a problem that led to the study of power domination in graphs. The power domination number of paths or cycles is one. As a result, a single-phase measuring unit (PMU) keeps track of all the vertices of a path or cycle in circuits. When the single connected PMU in a path or circle is not functioning effectively under unexpected conditions, then the observability of the electrical network diminishes. To overcome this problem, Double Power Domination was introduced. A subset 𝒮 of the vertex set of a graph G is called a double power dominating set if every vertex in G is monitored at least twice by the vertices in 𝒮. The minimal cardinality of all double power dominating sets 𝒮 is known as the double power domination number and is symbolized by γDP (G). In this article, we have proposed an algorithm to obtain the double power domination number for any graph. We have related the double power domination number and power domination number. We have demonstrated that the difference between the power domination number and the double power domination number is minimal.