AIJRSTEM19-

In 1962 ,Ore used the name " dominating set" and "domination number". In 1977 , Cockayne and Hedetniemi made an interesting and extensive survey of the results known at that time about dominating sets in graphs. The survey paper of cockayne and Hedetniemi has generated a lot of interest in the study of domination in graphs. Domination has a wide range of applications in Radio station, modeling social networks, coding theory, nuclear power plants problems. A non-empty subset D of vertices in a graph G=(V,E) is a "Dominating set", if every vertex in V-D is adjacent to some vertex in D. The domination numberγ(G) of G is the minimum cardinality of a minimal dominating set. A total dominating set Dt of G is a dominating set such that the induced subgraph <D> has no isolated vertices. The total domination number γt(G) of G is the minimum cardinality of a minimal total dominating set of G. Let D be the minimum dominating set of G. If V-D contains a dominating set say ′,then ′ is called an inverse dominating set with respect to D. The invese domination number ′( )of G is the order of the smallest inverse dominating set of G. "Kulli and Sigarkandi" introduced the concept of inverse domination in Graphs. Let G = (V,E) be a simple graph. Let Dt be the minimum total dominating set of G. If V-Dt contains a total dominating set ′ of G, then ′ is called an inverse total dominating set with respect to D. The inverse total domination number ′( ) of G is the minimum number of vertices in an inverse total dominating set of G. The concept of total domination is first introduced by Cockayne, Dawes and Hedetniemi in 1980. V.R. Kulli and RadhaRajamaniIyer introduced the concept of inverse total domination in graphs. In this paper, we found the total domination number and inverse total domination number of some special classes of graphs.

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 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.

International Journal of Combinatorics, 2012

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V − D is adjacent to at least one vertex in D. The domination number γ G is the minimum cardinality of a dominating set of G. A subset of V − D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ G is the minimum cardinality of the inverse dominating sets. characterized connected graphs G with γ G γ G n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ G γ G n − 1.

Given a graph G, a set D ⊂ V (G) is a dominating set of G if every vertex not in D is adjacent to at least one vertex of D. The domination number γ(G) is the minimum cardinality of a dominating set of G. If moreover, every vertex not in D is adjacent to exactly one vertex of D, then D is called a perfect dominating set of G. The perfect domination number γ 11 (G) is the minimum cardinality of a perfect dominating set of G. In general, for every integer k ≥ 1, a dominating set D is called a k-quasiperfect dominating set if every vertex not in D is adjacent to at most k vertices of D. The k-quasiperfect domination number γ 1k (G) is the minimum cardinality of a k-quasiperfect dominating set of G. These parameters are related in the following general way (Δ the maximum degree of G and by n the number of vertices): γ(G) = γ 1Δ (G) ≤ • • • ≤ γ 12 (G) ≤ γ 11 (G) ≤ n. In this work we study the perfect domination number, with the help of this decreasing chain of domination parameters, in the following graph families: graphs with extremal maximum degree, that is, graphs with Δ ≥ n − 3 or Δ = 3, and also in cographs, claw-free graphs and trees. We also study the behavior of these parameters under some usual product operations.

arXiv (Cornell University), 2015

The k-dominating graph D k (G) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k, two such sets being adjacent if they differ by either adding or deleting a single vertex. A graph is a dominating graph if it is isomorphic to D k (G) for some graph G and some positive integer k. Answering a question of Haas and Seyffarth for graphs without isolates, it is proved that if G is such a graph of order n ≥ 2 and with G ∼ = D k (G), then k = 2 and G = K 1,n-1 for some n ≥ 4. It is also proved that for a given r there exist only a finite number of r-regular, connected dominating graphs of connected graphs. In particular, C 6 and C 8 are the only dominating graphs in the class of cycles. Some results on the order of dominating graphs are also obtained.

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.

A domination in graphs is part of graph theory which has many applications. Its application includes the morphological analysis, computer network communication, social network theory, CCTV installation, and many others. A set D of vertices of a simple graph G, that is a graph without loops and multiple edges, is called a dominating set if every vertex u ∈ V (G) − D is adjacent to some vertex v ∈ D. The domination number of a graph G, denoted by γ(G), is the order of a smallest dominating set of G. A dominating set D with |D| = γ(G) is called a minimum dominating set, see Haynes and Henning [5]. This research aims to find the domination number of some families of special graphs, namely Spider Web graph W b n , Helmet graph H n,m , Parachute graph P c n , and any regular graph. The results shows that the resulting domination numbers meet the lower bound of an obtained lower bound γ(G) of any graphs.

arXiv (Cornell University), 2017

A set S ⊆ V is a dominating set of G if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) of G equals the minimum cardinality of a dominating set S in G; we say that such a set S is a γ-set. The single greatest focus of research in domination theory is the determination of the value of γ(G). By definition, all vertices must be dominated by a γ-set. In this paper we propose relaxing this requirement, by seeking sets of vertices that dominate a prescribed fraction of the vertices of a graph. We focus particular attention on 1/2 domination, that is, sets of vertices that dominate at least half of the vertices 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.