Total co-independent domination in graphs

Discrete Applied Mathematics, 2012

Let G = (V , E) be a graph with no isolated vertex. A subset of vertices S is a total dominating set if every vertex of G is adjacent to some vertex of S. For some α with 0 < α ≤ 1, a total dominating set S in G is an α-total dominating set if for every vertex v ∈ V \ S, |N(v) ∩ S| ≥ α|N(v)|. The minimum cardinality of an α-total dominating set of G is called the α-total domination number of G. In this paper, we study α-total domination in graphs. We obtain several results and bounds for the α-total domination number of a graph G.

2015

A subset S of vertices in a graph G is a global total dominating set, or just GTDS, if S is a total dominating set of both G and G. The global total domination number gt (G) of G is the minimum cardinality of a GTDS of G. In this paper, we show that the decision problem for gt (G) is NP-complete, and then characterize graphs G of order n with gt (G) = n 1.

IJCRT, 2019

Let G be a graph with minimal vertex dominating sets G1, G2,……..,Gm. Form a graph D(G) with vertices corresponding to G1, G2, ....,Gm and two sets Gi and Gj are adjacent if they have atleast one vertex in common. This graph D(G) is known as Dominating Graph. Minimum cardinality of a minimal dominating sets of D(G) is called domination number of D(G) and is denoted by γ(D(G)). In this paper, bounds on γ(D(G)) are obtained and its exact values for some standard graphs are found.

Discrete Mathematics

The k-restricted domination number of a graph G is the minimum number d k such that for any subset U of k vertices of G, there is a dominating set in G including U and having at most d k vertices. Some new upper bounds in terms of order and degrees for this number are found.

Iranian Journal of Science and Technology, Transactions A: Science

Let G = (V, E) be a graph. For some α with 0 < α ≤ 1, a subset S of V is said to be a α-partial dominating set if |N[S]| ≥ α|V |. The size of a smallest such S is called the αpartial domination number and is denoted by pd α (G). In this paper, we introduce α-partial domination number in a graph G and study different bounds on the partial domination number of a graph G with respect to its order, maximum degree, domination number etc., Moreover, α-partial domination spectrum is introduced and Nordhaus-Gaddum bounds on the partial domination number are studied.

Mathematica Slovaca, 2007

The k-restricted domination number of a graph G is the minimum number d k such that for any subset U of k vertices of G, there is a dominating set in G including U and having at most d k vertices. Some new upper bounds in terms of order and degrees for this number are found.

2008

Let G = (V,E) be a simple graph. A set S ⊆ V is a dominating set of graph G, if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set in G. It is well known that if e ∈ E(G), then γ(G−e)−1 ≤ γ(G) ≤ γ(G−e). In this paper, as an application of this inequality, we obtain the domination number of some certain graphs.

Discrete Mathematics, 1997

We are interested in a notion of domination related to both vertices and edges of graphs. We present results about Q(G), the total domination number of a graph G and study ad(G), which is the minimum value of IX*(S) where S is any total dominating set of G. In particular, we prove relations between these two parameters and give lower and upper bounds for Q(G).

2021

A set D of vertices in an isolate-free graph G is a semitotal dominating set of G if D is a dominating set of G and every vertex in D is within distance 2 from another vertex of D. The semitotal domination number of G is the minimum cardinality of a semitotal dominating set of G and is denoted by γt2(G). In this paper after computation of semitotal domination number of specific graphs, we count the number of this kind of dominating sets of arbitrary size in some graphs.

Results in Mathematics, 1990

A set S of vertices of a graph G is dominating if each vertex z not in S is adjacent to some vertex in S, and is independent if no two vertices in S are adjacent. The domination number,-y(G),