Total Partial Domination in Graphs Under Some Binary Operations

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...

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.

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.

Journal of Combinatorial Optimization, 2008

In this paper we continue the investigation of total domination in Cartesian products of graphs first studied in (Henning, M.A., Rall, D.F. in Graphs Comb. 21:63-69, 2005). A set S of vertices in a graph G is a total dominating set of G if every vertex in G is adjacent to some vertex in S. The maximum cardinality of a minimal total dominating set of G is the upper total domination number of G, denoted by Γ t (G). We prove that the product of the upper total domination numbers of any graphs G and H without isolated vertices is at most twice the upper total domination number of their Cartesian product; that is, Γ t (G)Γ t (H ) ≤ 2Γ t (G H ).

Graphs and Combinatorics, 2005

The most famous open problem involving domination in graphs is Vizing's conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this paper, we investigate a similar problem for total domination. In particular, we prove that the product of the total domination numbers of any nontrivial tree and any graph without isolated vertices is at most twice the total domination number of their Cartesian product, and we characterize the extremal graphs.

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.

Applied Mathematical Sciences, 2014

Let G = (V (G), E(G)) be a connected graph. A subset S of V (G) is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The set N G (v) is the set of all vertices of G adjacent to v. A subset S of V (G) is a locating set of G if N G (u) ∩ S = N G (v) ∩ S for every two distinct vertices u and v in V (G) \ S. A locating subset S of V (G) which is also a total dominating set is called a locating total dominating set of G. The minimum cardinality of a locating total dominating set of G is called the locating total domination number of G. In this paper, we characterize the locating total dominating sets in the Cartesian product of graphs. We also determine the relationships between the locating total domination number and some other domination parameters such as

Zenodo (CERN European Organization for Nuclear Research), 2017

A subset D of vertices of a connected graph G is called a semiglobal total dominating set if D is a dominating set for G and G sc and < D > has no isolated vertex in G, where G sc is the semi complementary graph of G. The semiglobal total domination number is the minimum cardinality of a semiglobal total dominating set of G and is denoted by γ sgt (G). In this paper exact values for γ sgt (G) are obtained for some graphs like cycles, wheel and paths are presented as well.

Applied Mathematical Sciences, 2014

A subset S of V (G), where G is a graph without isolated vertices, is a double dominating set of G if for each x ∈ V (G), |N G [x] ∩ S| ≥ 2. In this paper, we characterize the double dominating sets in the join, corona, and lexicographic product of two graphs. We also determine sharp bounds for the double domination numbers of these graphs. In particular, we show that if G and H are any connected non-trivial graphs, then γ dd (G [H]) ≤ min{2γ t (G), γ(G)γ dd (H)}, where γ, γ t , and γ dd are, respectively, the domination, total domination, and double domination parameters.

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.