How large can the domination numbers of a graph be?

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.

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.

International Journal of Pure and Apllied Mathematics

In this paper, we determine for a simple graph G on n vertices and m edges a variety of domination parameters such as connected domination number, outer connected domination number, doubly connected domination number, global domination number, total global connected domination number, 2-connected domination number, strong domination number, fair domination number, independence domination number etc.

Opuscula Mathematica

A set \(D\) of vertices of a graph \(G=(V_G,E_G)\) is a dominating set of \(G\) if every vertex in \(V_G-D\) is adjacent to at least one vertex in \(D\). The domination number (upper domination number, respectively) of \(G\), denoted by \(\gamma(G)\) (\(\Gamma(G)\), respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of \(G\). A subset \(D\subseteq V_G\) is called a certified dominating set of \(G\) if \(D\) is a dominating set of \(G\) and every vertex in \(D\) has either zero or at least two neighbors in \(V_G-D\). The cardinality of a smallest (largest minimal, respectively) certified dominating set of \(G\) is called the certified (upper certified, respectively) domination number of \(G\) and is denoted by \(\gamma_{\rm cer}(G)\) (\(\Gamma_{\rm cer}(G)\), respectively). In this paper relations between domination, upper domination, certified domination and upper certified domination numbers of a graph are studied.

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.

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.

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.

In a graph G = (V, E), S ⊆ V is a dominating set of G if every vertex is either in S or joined by an edge to some vertex in S. Many different types of domination have been researched extensively. This dissertation explores some new variations and applications of dominating sets.

Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, 2015

Let G = (V, E) be a graph. A subset D of V (G) is called a super dominating set if for every v ∈ V (G) − D there exists an external private neighbour of v with respect to V (G) − D. The minimum cardinality of a super dominating set is called the super domination number of G and is denoted by γsp(G). In this paper some results on the super domination number are obtained. We prove that if T is a tree with at least three vertices, then n 2 ≤ γsp(T) ≤ n − s, where s is the number of support vertices in T and we characterize the extremal trees.

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.