d-Lucky Labeling of Graphs

AbstractLet l: V (G) →N be a labeling of the vertices of a graph G by positive integers. Define , where d(u) denotes the degree of u and N(u) denotes the open neighborhood of u. In this paper we introduce a new labeling called d-lucky labeling and study the same as a vertex coloring problem. We define a labeling l as d-lucky if c(u) ≠ c(v) , for every pair of adjacent vertices u and v in G. The d-lucky number of a graph G, denoted by ηdl(G), is the least positive k such that G has a d-lucky labeling with {1,2, ..., k} as the set of labels. We obtain ηdl(G) = 2 for hypercube network, butterfly network, benes network, mesh network, hypertree and X-tree

2-Power Domination in Certain Interconnection Networks

AbstractThe k-power domination problem generalizes domination and power domination problems. The k-power domination problem is to determine a minimum size vertex set S ⊆ V(G) such that after setting X = N[S] and iteratively adding to X vertices x that have a neighbour v in X such that at most k neighbours of v are not yet in X till we get X = V(G). The least cardinality of such set is called the k-power domination number of G and is denoted by γp,k (G). In this paper, we restrict our discussion to k = 2, referred to as 2-power domination. We compute 2-power domination number for certain interconnection networks such as hypertree, sibling tree, X-tree, Christmas tree, mesh, honeycomb mesh, hexagonal mesh, cylinder, generalized Petersen graph and subdivision of graphs

Embeddings Between Hypercubes and Hypertrees

Abstract Graph embedding problems have gained importance in the field of interconnection networks for parallel computer architectures. Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. In this paper, we embed the rooted hypertree RHT (r) into r-dimensional hypercube Q r with dilation 2, r ≥ 2. Also, we compute the exact wirelength of the embedding of the r-dimensional hypercube Q r into the rooted hypertree RHT (r), r ≥ 2

2-power domination number for Knodel graphs and its application in communication networks

In a graph G, if each node v∈V (G)\S is connected to some node in S, then the set S of nodes is referred to as a dominating set. The domination number of G is the minimum cardinality of all dominating sets of G and is represented by γ(G). If a dominating set S monitors every node in the system under a set of guidelines for power systems monitoring, then the set S is referred to as a power-dominating set of G. The power domination number of G is the least number of vertices of a power dominating set of G. A generalization of power domination is the k-power domination in a graph G. The k-power domination number of G is the minimum cardinality of all k-power dominating sets of G and is represented by γp,k(G). In this paper, we have obtained the 2-power domination number represented by γp,2(G) for 4-regular Kn¨odel graphs and given the lower bound for 5-regular Kn¨odel graphs