Domination Number Of Product Research Articles

On domination number of Cartesian product of directed paths

Let ?(G) denote the domination number of a digraph G and let P m ?P n denote the Cartesian product of P m and P n , the directed paths of length m and n. In this paper, we give a lower and upper bound for ?(P m ?P n ). Furthermore, we obtain a necessary and sufficient condition for P m ?P n to have efficient dominating set, and determine the exact values: ?(P 2?P n )=n, $\gamma(P_{3}\square P_{n})=n+\lceil\frac{n}{4}\rceil$ , $\gamma(P_{4}\square P_{n})=n+\lceil\frac{2n}{3}\rceil$ , ?(P 5?P n )=2n+1 and $\gamma(P_{6}\square P_{n})=2n+\lceil\frac{n+2}{3}\rceil$ .

On domination number of Cartesian product of directed cycles

Let γ ( G ) denote the domination number of a digraph G and let C m □ C n denote the Cartesian product of C m and C n , the directed cycles of length m , n ⩾ 2 . In this paper, we determine the exact values: γ ( C 2 □ C n ) = n ; γ ( C 3 □ C n ) = n if n ≡ 0 ( mod 3 ) , otherwise, γ ( C 3 □ C n ) = n + 1 ; γ ( C 4 □ C n ) = 3 n 2 if n ≡ 0 ( mod 8 ) , otherwise, γ ( C 4 □ C n ) = n + ⌈ n + 1 2 ⌉ .

On the [formula omitted]-domination number of Cartesian products of graphs

Let G □ H denote the Cartesian product of graphs G and H . In this paper, we study the { k } -domination number of Cartesian product of graphs and give a new lower bound of γ { k } ( G □ H ) in terms of packing and { k } -domination numbers of G and H . As applications of this lower bound, we prove that: (i) For k = 1 , the new lower bound improves the bound given by Chen, et al. [G. Chen, W. Piotrowski, W. Shreve, A partition approach to Vizing’s conjecture, J. Graph Theory 21 (1996) 103–111]. (ii) The product of the { k } -domination numbers of two any graphs G and H , at least one of which is a ( ρ , γ ) -graph, is no more than k γ { k } ( G □ H ) . (iii) The product of the { 2 } -domination numbers of any graphs G and H , at least one of which is a ( ρ , γ − 1 ) -graph, is no more than 2 γ { 2 } ( G □ H ) .

On the total {k}-domination number of Cartesian products of graphs

Let γ t {k} (G) denote the total {k}-domination number of graph G, and let \(G\mathbin{\square}H\) denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices, \(\gamma _{t}^{\{k\}}(G)\gamma _{t}^{\{k\}}(H)\le k(k+1)\gamma _{t}^{\{k\}}(G\mathbin{\square}H)\) . As a corollary of this result, we have \(\gamma _{t}(G)\gamma _{t}(H)\le 2\gamma _{t}(G\mathbin{\square}H)\) for all graphs G and H without isolated vertices, which is given by Pak Tung Ho (Util. Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63–69, 2005).

On the upper total domination number of Cartesian products of graphs

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 Γt(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, Γt(G)Γt(H)≤2Γt(G □ H).

On the Total Domination Number of Cartesian Products of Graphs

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.

A note on domination parameters of the conjunction of two special graphs

A dominating set D of G is called a split dominating set of G if the subgraph induced by the subset V (G) i D is disconnected. The cardinality of a minimum split dominating set is called the minimum split domination number of G: Such subset and such number was introduced in [4]. In [2], [3] the authors estimated the domination number of products of graphs. More precisely, they were study products of paths. Inspired by those results we give another estimation of the domination number of the conjunction (the cross product) Pn ^G: The split domination number of Pn ^G also is determined. To estimate this number we use the minimum connected domination number ∞c(G):

Some results on domination number of products of graphs

Let G= (V,E) be a simple graph. A subset D of V is called a dominating set of G if for every vertex κ,eV—D,κ is adjacent to at least one vertex of D. Let γ(G) and γc(G) denote the domination and connected domination number of G, respectively. In 1965,Vizing conjectured that if GXH is the Cartesian product of G and H, then $$\gamma (G \times H) \geqslant \gamma (G) \cdot \gamma (H).$$ . In this paper, it is showed that the conjecture holds if Y(H)=#γc(H). And for paths Pm and Pn, a lower bound and an upper bound for γ(PmXPn) are obtained.