For a graph G , we define the lower bipartite number LB( ) as minimum order of maximal induced subgraph . We study parameter, and related parameter domination, providing bounds both in general graphs some families. example, show that there are arbitrarily large 4-connected planar with = 4 but 5-connected has linear ). also if is outerplanar n then lies between ( + 2)/3 2 /3, these sharp.

Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measurement devices placed on a set S of vertices of a graph G, the set of monitored vertices is initially the set S together with all its neighbors. The...

In this paper, we investigate domination number as well as signed domination numbers of Cay(G : S) for all cyclic group G of order n, where n in {p^m; pq} and S = { a^i : i in B(1; n)}. We also introduce some families of connected regular graphs gamma such that gamma_S(Gamma) in {2,3,4,5 }.

Upper and lower bounds on the total domination number of the direct product of graphs are given. The bounds involve the {2}-total domination number, the total 2-tuple domination number, and the open packing number of the factors. Using these relationships one exact total domination number is obtained. An infinite family of graphs is constructed showing that the bounds are best possible. The dom...

Let G be a simple graph, and let p be a positive integer. A subset D ⊆ V (G) is a p-dominating set of the graph G, if every vertex v ∈ V (G)−D is adjacent with at least p vertices of D. The p-domination number γp(G) is the minimum cardinality among the p-dominating sets of G. Note that the 1-domination number γ1(G) is the usual domination number γ(G). If G is a nontrivial connected block graph,...

Abstract: Let G = (VG, EG) be a simple connected graph. The eccentric distance sum of G is defined as ξ(G) = ∑ v∈VG εG(v)DG(v), where εG(v) is the eccentricity of the vertex v and DG(v) = ∑ u∈VG dG(u, v) is the sum of all distances from the vertex v. In this paper the tree among n-vertex trees with domination number γ having the minimal eccentric distance sum is determined and the tree among n-...

A vertex of a graph is said to dominate itself and all of its neighbors. A double dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D. The double domination number of a graph G is the minimum cardinality of a double dominating set of G. For a graph G = (V,E), a subset D ⊆ V (G) is a 2dominating set if every vertex of V (...