Domination in graphs has been extensively studied and adopted in many real life applications. The monitoring electrical power system is a variant of a domination problem called power domination problem. Another variant is the zero forcing problem. Determining minimum cardinality of a power dominating set and zero forcing set in a graph are the power domination problem and zero forcing problem, respectively. Both problems are $NP$-complete. In this paper, we compute the power domination number and the zero forcing number for fully connected cubic networks.

Finding a minimum-sized set of vertices in a graph such that each edge in the graph is incident to at least one vertex in the set is the objective of the study of vertex cover problem. Mathematically, a vertex cover of a graph G is a set S of vertices in G such that every edge of G has at least one end vertex in S. The vertex covering number denoted by β(G) is the minimum cardinality taken over all vertex covering sets of G. In this paper, we obtain the vertex cover number of certain architectures such as hypertrees, glued trees, triangular graphs, and honeycomb rectangular torus.

The problem of monitoring an electric power system by placing Phasor Measurement Units in the system as possible is closely related to the well-known power domination problem in graphs. In this paper, we compute, 2-power domination number for fully connected cubic networks and also developed k-power domination algorithm for specific networks and evaluates its performance across these networks. Furthermore, the algorithm’s outputs are used for various other applications such as fault detection and localization, state identification, accuracy improvement, and cyber security enhancement, thereby making it a valuable tool for utilities and grid operators.

A set S of black colored vertices in a graph G is called a zero forcing set of G if at every discrete time step, a black vertex with exactly one non-black vertex as its neighbour forces it to be colored black and by iteratively applying the procedure, all vertices in G are colored black. If S induces an independent set of edges, then S is termed an edge-forcing set of G. The minimum cardinality of edge-forcing sets of G, denoted by ζe (G) is called edge-forcing number of G. In this paper, we obtain the edge-forcing number for Sierpiński graphs and Sierpiński gasket graphs.