From the Quasi-Total Strong Differential to Quasi-Total Italian Domination in Graphs (original) (raw)

Neighbourhood total domination in graphs

Opuscula Mathematica, 2011

Let G = (V, E) be a graph without isolated vertices. A dominating set S of G is called a neighbourhood total dominating set (ntd-set) if the induced subgraph N (S) has no isolated vertices. The minimum cardinality of a ntd-set of G is called the neighbourhood total domination number of G and is denoted by γnt(G). The maximum order of a partition of V into ntd-sets is called the neighbourhood total domatic number of G and is denoted by dnt(G). In this paper we initiate a study of these parameters.

The differential and the Roman domination number of a graph

Let G = (V, E) be a graph of order n and let B(S) be the set of vertices in V \ S that have a neighbor in the vertex set S. The differential of a vertex set S is defined as ∂(S) = |B(S)| − |S| and the maximum value of ∂(S) for any subset S of V is the differential of G. A Roman dominating function of G is a function f : V → {0, 1, 2} such that every vertex u with f (u) = 0 is adjacent to a vertex v with f (v) = 2. The weight of a Roman dominating function is the value f (V ) = u∈V f (u). The minimum weight of a Roman dominating function of a graph G is the Roman domination number of G, written γR(G). We prove that γR(G) = n − ∂(G) and present several combinatorial, algorithmic and complexity-theoretic consequences thereof.

Differentiating total domination in graphs: revisited

International Journal of Mathematical Analysis, 2014

Let G = (V (G), E(G)) be a connected graph. A subset S of V (G) is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The set N G (v) is the set of all vertices of G adjacent to v. A subset S of V (G) is a differentiating set of G if N G [u] ∩ S = N G [v] ∩ S for every two distinct vertices u and v in V (G). A differentiating subset S of V (G) which is also a total dominating is called a differentiating total dominating set of G. The minimum cardinality of a differentiating total dominating set of G is called the differentiating total domination number of G. In this paper, we characterize the differentiating total dominating sets in the Cartesian product of graphs. We also determine the relationships between the differentiating total domination number and some other domination number such as domination number, differentiating domination number, and locating total domination number.

Domination number and neighbourhood conditions

Discrete Mathematics, 1999

The domination number 7 of a graph G is the minimum cardinality of a subset D of vertices of G such that each vertex outside D is adjacent to at least one vertex in D. For any subset A of the vertex set of G, let O+(A) be the set of vertices not in A which are adjacent to at least one vertex in A. Let N(A) be the union of A and 0+(A), and d(A) be the sum of degrees of all the vertices of A. In this paper we prove the inequality 2q ~<(p-y)(p-y+2)-Id+(A)l (p-y+l)+d(N(A)), and characterize the extremal graphs for which the equality holds, where p and q are the numbers of vertices and edges of G, respectively. From this we then get an upper bound for y which generalizes the known upper bound 7 ~< P + 1-x/~ + 1. Let I(A) be the set of vertices adjacent to all vertices of A, and i(A) be the union of A and I(A). We prove that 2q<<.(p-7-]i(A)] + 2)(p-~ + 4) + d(i(A))-min{p-~,-[I(A)] + 2, ]AI, II(A)I, 3}, which implies an upper bound for y as well. (~) 1999 Elsevier Science B.V. All rights reserved

Neighborhood Total 2-Domination in Graphs

2017

Let G = (V,E) be a graph without isolated vertices. A set S ⊆ V is called the neighborhood total 2-dominating set (nt2d-set) of a graph G if every vertex in V − S is adjacent to at least two vertices in S and the induced subgraph < N(S) > has no isolated vertices. The minimum cardinality of a nt2d-set of G is called the neighborhood total 2domination number of G and is denoted by γ2nt(G). In this paper we initiate a study of this parameter.

A note on neighborhood total domination in graphs

Proceedings - Mathematical Sciences, 2015

Let G = (V , E) be a graph without isolated vertices. A dominating set S of G is called a neighborhood total dominating set (or just NTDS) if the induced subgraph G[N(S)] has no isolated vertex. The minimum cardinality of a NTDS of G is called the neighborhood total domination number of G and is denoted by γ nt (G).In this paper, we obtain sharp bounds for the neighborhood total domination number of a tree. We also prove that the neighborhood total domination number is equal to the domination number in several classes of graphs including grid graphs.

Weakly connected total domination in graphs

International Mathematical Forum, 2016

Let G = (V (G), E(G)) be a connected undirected graph. The closed neighborhood of any vertex v ∈ V (G) is N G [v] = {u ∈ V (G) : uv ∈ E(G)} ∪ {v}. For C ⊆ V (G), the closed neighborhood of C is N [C] = ∪ v∈C N G [v]. A set S ⊆ V (G) is a total dominating set of G if for each x ∈ V (G), there exists y ∈ S such that xy ∈ E(G), that is, N (S) = V (G). A total dominating set S ⊆ V (G) is a weakly connected total dominating set of a connected graph G if the subgraph S w = (N G (S), E w) weakly induced by S is connected, where E w is the set of all edges with at least one vertex in S. The weakly connected total domination number of G, denoted by γ wt (G), is the minimum cardinality among all weakly connected total dominating sets of G. In this paper, the weakly connected total dominating sets in graphs resulting from some binary operations are characterized. As consequences, the weakly connected total domination number of the aforementioned graphs are determined.