Total mixed domination in graphs

2018

For a graph G = (V,E), we call a subset S ⊆ V ∪ E a total mixed dominating set of G if each element of V ∪ E is either adjacent or incident to an element of S, and the total mixed domination number γtm(G) of G is the minimum cardinality of a total mixed dominating set of G. In this paper, we initiate to study the total mixed domination number of a connected graph by giving some tight bounds in terms of some parameters such as order and total domination numbers of the graph and its line graph. Then we discuss on the relation between total mixed domination number of a graph and its diameter. Studing of this number in trees is our next work. Also we show that the total mixed domination number of a graph is equale to the total domination number of a graph which is obtained by the graph. Giving the total mixed domination numbers of some special graphs is our last work.

Discrete Mathematics, 2004

A dominating set for a graph G = (V; E) is a subset of vertices D ⊆ V such that for all v ∈ V − D there exists some u ∈ D adjacent to v. The domination number of G is the size of its smallest dominating set. A dominating set D is a total dominating set if every vertex in D has a neighbor in D. We give a tight upper bound on the number of edges that a connected graph with a given total domination number can have, and characterize the extremal graphs attaining the bound. We do the same for the k-restricted domination number, which is the smallest number d, such that for any subset U ⊆ V where |U | = k there exists a dominating set for G of size at most d, and containing all vertices in U .

European Journal of Pure and Applied Mathematics, 2021

This paper introduces and investigates a variant of partial domination called the connected α-partial domination. For any graph G = (V (G), E(G)) and α ∈ (0, 1], a set S ⊆ V (G) is an α-partial dominating set in G if |N[S]| ≥ α |V (G)|. An α-partial dominating set S ⊆ V (G) is a connected α-partial dominating set in G if ⟨S⟩, the subgraph induced by S, is connected. The connected α-partial domination number of G, denoted by ∂Cα(G), is the smallest cardinality of a connected α-partial dominating set in G. In this paper, we characterize the connected α-partial dominating sets in the join and lexicographic product of graphs for any α ∈ (0, 1] and determine the corresponding connected α-partial domination numbers of graphs resulting from the said binary operations. Moreover, we establish sharp bounds for the connected α-partial domination numbers of the corona and Cartesian product of graphs. Furthermore, we determine ∂Cα(G) of some special graphs when α...

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 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.

Journal of Discrete Mathematical Sciences and Cryptography, 2009

A dominating set D of a graph G is a split dominating set if the induced subgraph < V − D > is disconnected. The split domination number γ s (G) is the minimum cardinality of a split dominating set. The concept of split domination number was introduced by Kulli and Janakiram. In this paper, some results on split domination are obtained.

Graphs and Combinatorics, 2013

Let G be a graph with vertex set V . A set D ⊆ V is a total restrained dominating set of G if every vertex in V has a neighbor in D and every vertex in V \ D has a neighbor in V \ D. The minimum cardinality of a total restrained dominating set of G is called the total restrained domination number of G, and is denoted by γ tr (G). In this paper, we prove that if G is a connected graph of order n ≥ 4 and minimum degree at least two, then γ tr (G) ≤ n − 3 √ n 4 .

Discrete Mathematics, 2000

A dominating set for a graph G = (V; E) is a subset of vertices V ⊆ V such that for all v ∈ V − V there exists some u ∈ V adjacent to v. The domination number of G, denoted by (G), is the size of its smallest dominating set(s). When G is connected, we say V is a connected dominating set if the subgraph of G induced by V is connected. The connected domination number of G is the size of its smallest connected dominating set, and is denoted by c(G). In this paper we determine the maximum number of edges that a connected graph with a given number of vertices and a given connected domination number can have. We also characterize the extremal graphs achieving the bound.

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.

Central European Journal of Mathematics, 2010

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Karami, Khoeilar, Sheikholeslami and Khodkar, (Graphs and Combinatorics, 2009, 25, 727–733) proved that for any connected graph G of order n ≥ 3, sdγt (G) ≤ 2γ t (G) − 1 and posed the following problem: Characterize the graphs that achieve the aforementioned upper bound. In this paper we first prove that sdγt (G) ≤ 2α′(G) for every connected graph G of order n ≥ 3 and δ(G) ≥ 2 where α′(G) is the maximum number of edges in a matching in G and then we characterize all connected graphs G with sdγt (G)=2γt (G)−1.