A Note on “on The Power Dominating Sets of Hypercubes”

1. Any vertex that is incident to an observed edge is observed. 2. Any edge joining two observed vertices is observed. The power domination problem is a variant of the classical domination problem in graphs and is defined as follows. Given an undirected graph G = (V, E), the problem is to find a minimum vertex set SP ⊆ V , called the power dominating set of G, such that all vertices in G are observed by the vertices of Sp. Herein, a vertex observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. The minimum cardinality of a power dominating set of a graph is its power domination number. In 2011, Dean et al gave the lower bound and upper bound of power domination number on hypercubes. In this paper, we present the known values of power domination number on Hypercubes and disprove Dean’s conjecture. 3. If a vertex is incident to a total of k >1 edges and if k 1 of these edges are observed, then all k of these edges are observed. Haynes et al. [10] first considered the graph theoretical representation of the power system monitoring problem as a variation of the well-known graph domination problem (see also [11, 12]). A set S ⊆ V is a dominating set in G if every vertex in V S has at least one neighbor in S. The cardinality of a minimum dominating set of G is the domination number γ (G). Considering the power system monitoring problem as a variation of the dominating set problem, let a set Sp be a power dominating set (abbreviated PDS) if every vertex and every edge in G is observed by Sp . The power domination number γp(G) is the minimum cardinality of a PDS of G. A PDS of G with the minimum cardinality is called a γp(G)-set. Since any dominating set is a power dominating set, 1 ≤ γp(G) ≤ γ(G) for all graphs G. Haynes et al. [10] also showed that the power domination problem is NP-complete even when restricted to some special classed of graphs such as bipartite graphs or chordal graphs. The power domination problem is widely studied in [1-8, 10, 13, 15-25].