On the doubly connected domination number of a graph

Let G be a connected simple graph. A set S ⊆ V (G) is a doubly connected dominating set if it is dominating and both S and V (G)\S are connected. The doubly connected domination number of G, denoted by γ cc (G), is the smallest cardinality of a doubly connected dominating set S of G. A convex dominating set S of G is a convex doubly connected dominating set if S is a doubly connected dominating set of G. The convex doubly connected domination number of G, denoted by γ ccc (G), is the smallest cardinality of a convex doubly connected dominating set S of G. In this paper, we show that every integers a, b, c, and n with 1 ≤ a ≤ b ≤ c ≤ n is realizable as domination number, convex domination number, convex doubly connected domination number, and order of G respectively. Further, we give the characterization of the convex doubly connected dominating set with convex doubly connected domination numbers of 1 and 2. Finally, we characterize the convex doubly connected dominating sets of the join and corona of graphs.

In a graph G, a vertex dominates itself and its neighbours. A subset S of V is called a dominating set in G if every vertex in V is dominated by at least one vertex in S. The domination number  (G) is the minimum cardinality of a dominating set. A set S V is called a double dominating set of a graph G if every vertex in V is dominated by at least two vertices in S. The minimum cardinality of a double dominating set is called double domination number of G and is denoted by dd(G). The connectivity  (G) of a connected graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. In this paper we find an upper bound for the sum of the double domination number and connectivity of a graph and characterize the corresponding extremal graphs.

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.

Let G be a simple connected graph. A connected dominating set S ⊆ V (G) is called a doubly connected dominating set of G if the subgraph V (G)\S induced by V (G)\S is connected. In this paper, we show that any three positive integers a, b, and c with 4

Discrete Applied Mathematics, 2010

A set S of vertices of a connected graph G is a doubly connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraphs induced by S and V − S are connected. The doubly connected domination number γ cc (G) is the minimum size of such a set. We prove that when G and G are both connected of order n, γ cc (G) + γ cc (G) ≤ n + 3 and we describe the two infinite families of extremal graphs achieving the bound.

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...

International Journal of Mathematics and Soft Computing, 2015

A subset S of V is called a dominating set in G if every vertex in V -S is adjacent to at least one vertex in S. A dominating set S is said to be a restrained dominating set if V -S contains no isolated vertices. The minimum cardinality of a restrained dominating set of G is called the restrained domination number of G and is denoted by γ r (G). The connectivity κ(G) of a graph G is the minimum number of vertices whose removal results in a disconnected or trivial graph. In this paper we characterized the graphs with sum of restrained domination number and connectivity is equal to 2n -6.

Discrete Applied Mathematics, 2013

A subset S of vertices in a graph G = (V , E) is a connected dominating set of G if every vertex of V \S is adjacent to a vertex in S and the subgraph induced by S is connected. The minimum cardinality of a connected dominating set of G is the connected domination number γ c (G). The girth g(G) is the length of a shortest cycle in G. We show that if G is a connected graph that contains at least one cycle, then γ c (G) ≥ g(G) − 2, and we characterize the graphs obtaining equality in this bound. We also establish various upper bounds on the connected domination number of a graph, as well as Nordhaus-Gaddum type results.

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.

Let G(V, E) be a simple, finite, undirected connected graph. A non-empty set S ⊆ V of a graph G is a dominating set, if every vertex in V − S is adjacent to atleast one vertex in S. A dominating set S ⊆ V is called a locating dominating set, if for any two vertices v, w ∈ V − S, N (v) ∩ S = N (w) ∩ S. A locating dominating set S ⊆ V is called a co-isolated locating dominating set, if there exists atleast one isolated vertex in < V − S >. The co-isolated locating domination number γ cild is the minimum cardinality of a co-isolated locating dominating set. A graph G is called doubly-connected if both G and its complement G are connected.