Domination Number Of Product Research Articles

The 2-rainbow domination number of Cartesian product of cycles

The 2-rainbow domination number of Cartesian product of cycles

Domination number of Cartesian product through space projections

Domination number of Cartesian product through space projections

Double total domination number of Cartesian product of paths

<abstract><p>A vertex set $ S $ of a graph $ G $ is called a double total dominating set if every vertex in $ G $ has at least two adjacent vertices in $ S $. The double total domination number $ \gamma_{\times 2, t}(G) $ of $ G $ is the minimum cardinality over all the double total dominating sets in $ G $. Let $ G \square H $ denote the Cartesian product of graphs $ G $ and $ H $. In this paper, the double total domination number of Cartesian product of paths is discussed. We determine the values of $ \gamma_{\times 2, t}(P_i\square P_n) $ for $ i = 2, 3 $, and give lower and upper bounds of $ \gamma_{\times 2, t}(P_i\square P_n) $ for $ i \geq 4 $.</p></abstract>

Tadpole domination number of graphs

The graph obtained by joining cycle [Formula: see text] to a path [Formula: see text] with a bridge is called Tadpole graph denoted by [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is said to be a tadpole dominating set of [Formula: see text] if [Formula: see text] is a dominating set and the set of vertices of [Formula: see text] spans a tadpole graph [Formula: see text] where [Formula: see text], [Formula: see text]. In this paper, we find the tadpole domination number of cartesian product of certain graphs, namely, paths, cycles and complete graphs. Also the tadpole domination number for the Mycielskian of cycles and paths is obtained.

The secure domination number of Cartesian products of small graphs with paths and cycles

The secure domination numbers of the Cartesian products of two small graphs with paths or cycles is determined, as well as for Möbius ladder graphs. Prior to this work, in all cases where the secure domination number has been determined, the proof has either been trivial, or has been derived from lower bounds established by considering different forms of domination. However, the latter mode of proof is not applicable for most graphs, including those considered here. Hence, this work represents the first attempt to determine secure domination numbers via the properties of secure domination itself, and it is expected that these methods may be used to determine further results in the future.

More Results on Italian Domination in Cn□Cm

Italian domination can be described such that in an empire all cities/vertices should be stationed with at most two troops. Every city having no troops must be adjacent to at least two cities with one troop or at least one city with two troops. In such an assignment, the minimum number of troops is the Italian domination number of the empire/graph is denoted as γ I . Determining the Italian domination number of a graph is a very popular topic. Li et al. obtained γ I ( C n □ C 3 ) and γ I ( C n □ C 4 ) (weak {2}-domination number of Cartesian products of cycles, J. Comb. Optim. 35 (2018): 75–85). Stȩpień et al. obtained γ I ( C n □ C 5 ) = 2 n (2-Rainbow domination number of C n □ C 5 , Discret. Appl. Math. 170 (2014): 113–116). In this paper, we study the Italian domination number of the Cartesian products of cycles C n □ C m for m ≥ 6 . For n ≡ 0 ( mod 3 ) , m ≡ 0 ( mod 3 ) , we obtain γ I ( C n □ C m ) = m n 3 . For other C n □ C m , we present a bound of γ I ( C n □ C m ) . Since for n = 6 k , m = 3 l or n = 3 k , m = 6 l ( k , l ≥ 1 ) , γ r 2 ( C n □ C m ) = m n 3 , (the Cartesian product of cycles with small 2-rainbow domination number, J. Comb. Optim. 30 (2015): 668–674), it follows in this case that C n □ C m is an example of a graph class for which γ I = γ r 2 , which can partially answer the question presented by Brešar et al. on the 2-rainbow domination in graphs, Discret. Appl. Math. 155 (2007): 2394–2400.

The Signed Domination Number of Cartesian Product of Two Paths

Let G be a finite connected simple graph with vertex set V(G) and edge set E(G). A function f:V(G) → {1,1} is a signed dominating function if for every vertex v∈V(G), the closed neighborhood of v contains more vertices with function values 1 than with −1. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. In this paper, we calculate The signed domination numbers of the Cartesian product of two paths Pm and Pn for m = 3, 4, 5 and arbitrary n.

Connected domination game played on Cartesian products

Abstract The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of G. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the connected game domination number of G. Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P3 with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with Kk for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.

On the outer-connected domination number for graph products

An outer-connected dominating set for an arbitrary graph G is a set D̃ ⊆ V such that D̃ is a dominating set and the induced subgraph G[V\\D̃] be connected. In this paper, we focus on the outer-connected domination number of the product of graphs. We investigate the outer-connected dominating set in the lexicographic product of two arbitrary graphs, and we present upper bounds for outer-connected domination number in the lexicographic and the Cartesian product of graphs. We establish an equivalent form of the Vizing’s conjecture for outer-connected domination number in the lexicographic and the Cartesian product as . We also show that the outer-connected domination number of the Corona product of a connected graph G of order n and some non-trivial graph H equals n times the domination number of H. Furthermore, we study the outer-connected domination number of the direct product of finitely many complete graphs.

More Results on the Domination Number of Cartesian Product of Two Directed Cycles

Let γ ( D ) denote the domination number of a digraph D and let C m □ C n denote the Cartesian product of C m and C n , the directed cycles of length n ≥ m ≥ 3 . Liu et al. obtained the exact values of γ ( C m □ C n ) for m up to 6 [Domination number of Cartesian products of directed cycles, Inform. Process. Lett. 111 (2010) 36–39]. Shao et al. determined the exact values of γ ( C m □ C n ) for m = 6 , 7 [On the domination number of Cartesian product of two directed cycles, Journal of Applied Mathematics, Volume 2013, Article ID 619695]. Mollard obtained the exact values of γ ( C m □ C n ) for m = 3 k + 2 [M. Mollard, On domination of Cartesian product of directed cycles: Results for certain equivalence classes of lengths, Discuss. Math. Graph Theory 33(2) (2013) 387–394.]. In this paper, we extend the current known results on C m □ C n with m up to 21. Moreover, the exact values of γ ( C n □ C n ) with n up to 31 are determined.

Bagging Approach for Italian Domination in $C_{n} \square\,P_{m}$

For graph G = (V, E), the open neighborhood of a vertex v is N(v) = {u ∈ V|uv ∈ E}. A function f : V (G) → {0, 1, 2} is called an Italian dominating function of G if Σ u∈N(v) f (u) ≥ 2, for each vertex f (v) = 0. The weight of f is w( f) = Σ v∈V(G) f (v). The minimum weight of an Italian dominating function of G is called the Italian domination number of G, denoted by y I (G). In this paper, we present a bagging approach and a partitioning approach to investigate the Italian domination number of Cartesian product of circles and paths, Cn□ P m . We determine the exact values of the Italian domination numbers of C n □ P 3 , C 3 □ P m . We also present some bounds on the Italian domination number of C n □ P m for n, m ≥ 4.

Complementary tree nil domination number of Cartesian Product of Graphs

Complementary tree nil domination number of Cartesian Product of Graphs

Weak {2}-domination number of Cartesian products of cycles

For a graph $$G=(V, E)$$ , a weak $$\{2\}$$ -dominating function $$f:V\rightarrow \{0,1,2\}$$ has the property that $$\sum _{u\in N(v)}f(u)\ge 2$$ for every vertex $$v\in V$$ with $$f(v)= 0$$ , where N(v) is the set of neighbors of v in G. The weight of a weak $$\{2\}$$ -dominating function f is the sum $$\sum _{v\in V}f(v)$$ and the minimum weight of a weak $$\{2\}$$ -dominating function is the weak $$\{2\}$$ -domination number. In this paper, we introduce a discharging approach and provide a short proof for the lower bound of the weak $$\{2\}$$ -domination number of $$C_n \Box C_5$$ , which was obtained by Stȩpien, et al. (Discrete Appl Math 170:113–116, 2014). Moreover, we obtain the weak $$\{2\}$$ -domination numbers of $$C_n \Box C_3$$ and $$C_n \Box C_4$$ .

Total and paired domination numbers of $$C_m$$ C m bundles over a cycle $$C_n$$ C n

Let $$G=(V,E)$$G=(V,E) be a simple graph without isolated vertices. A set $$S$$S of vertices is a total dominating set of a graph $$G$$G if every vertex of $$G$$G is adjacent to some vertex in $$S$$S. A paired dominating set of $$G$$G is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired dominating set) is the total domination number (respectively, the paired domination number). Hu and Xu (J Combin Optim 27(2):369---378, 2014) computed the exact values of total and paired domination numbers of Cartesian product $$C_n\square C_m$$CnźCm for $$m=3,4$$m=3,4. Graph bundles generalize the notions of covering graphs and Cartesian products. In this paper, we generalize these results given in Hu and Xu (J Combin Optim 27(2):369---378, 2014) to graph bundle and compute the total domination number and the paired domination number of $$C_m$$Cm bundles over a cycle $$C_n$$Cn for $$m=3,4$$m=3,4. Moreover, we give the exact value for the total domination number of Cartesian product $$C_n\square C_5$$CnźC5 and some upper bounds of $$C_m$$Cm bundles over a cycle $$C_n$$Cn where $$m\ge 5$$mź5.

On Clark and Suen bound-type results for k-domination and Roman domination of Cartesian product graphs

A set S of vertices of a graph G is a dominating set for G if every vertex of G is adjacent to at least one vertex of S. The domination number γ(G), of G, is the minimum cardinality of a dominating set in G. Moreover, if the maximum degree of G is Δ, then for every positive integer k≤Δ, the set S is a k-dominating set in G if every vertex outside of S is adjacent to at least k vertices of S. The k-domination number of G, denoted by γ k (G), is the minimum cardinality of a k-dominating set in G. A map f: V→<texlscub>0, 1, 2</texlscub>is a Roman dominating function for G if for every vertex v with f(v)=0, there exists a vertex u∈N(v) such that f(u)=2. The weight of a Roman dominating function is f(V)=∑ u∈V f(u). The Roman domination number γR(G), of G, is the minimum weight of a Roman dominating function on G. In this paper, we obtain that for any two graphs G and H, the k-domination number of the Cartesian product of G and H is bounded below by γ(G)γ k (H)/2. Also, we obtain that the domination number of Cartesian product of G and H is bounded below by γ(G)γR(H)/3.

On the Domination Number of Cartesian Product of Two Directed Cycles

Denote byγ(G)the domination number of a digraphGandCm□Cnthe Cartesian product ofCmandCn, the directed cycles of lengthm,n≥2. In 2010, Liu et al. determined the exact values ofγ(Cm□Cn)form=2,3,4,5,6. In 2013, Mollard determined the exact values ofγ(Cm□Cn)form=3k+2. In this paper, we give lower and upper bounds ofγ(Cm□Cn)withm=3k+1for different cases. In particular,⌈2k+1n/2⌉≤γ(C3k+1□Cn)≤⌊2k+1n/2⌋+k. Based on the established result, the exact values ofγ(Cm□Cn)are determined form=7and 10 by the combination of the dynamic algorithm, and an upper bound forγ(C13□Cn)is provided.

Roman domination in Cartesian product graphs and strong product graphs

A map f : V ? {0, 1, 2} is a Roman dominating function for G if for every vertex v with f(v) = 0, there exists a vertex u, adjacent to v, with f(u) = 2. The weight of a Roman dominating function is f(V ) = ?u?v f(u). The minimum weight of a Roman dominating function on G is the Roman domination number of G. In this article we study the Roman domination number of Cartesian product graphs and strong product graphs.

2-Rainbow domination number of Cartesian products: $$C_{n}\square C_{3}$$ and $$C_{n}\square C_{5}$$

A function $$f:V(G)\rightarrow \mathcal P (\{1,\ldots ,k\})$$ is called a $$k$$ -rainbow dominating function of $$G$$ (for short $$kRDF$$ of $$G)$$ if $$ \bigcup \nolimits _{u\in N(v)}f(u)=\{1,\ldots ,k\},$$ for each vertex $$ v\in V(G)$$ with $$f(v)=\varnothing .$$ By $$w(f)$$ we mean $$\sum _{v\in V(G)}\left|f(v)\right|$$ and we call it the weight of $$f$$ in $$G.$$ The minimum weight of a $$ kRDF$$ of $$G$$ is called the $$k$$ -rainbow domination number of $$G$$ and it is denoted by $$\gamma _{rk}(G).$$ We investigate the $$2$$ -rainbow domination number of Cartesian products of cycles. We give the exact value of the $$2$$ -rainbow domination number of $$C_{n}\square C_{3}$$ and we give the estimation of this number with respect to $$C_{n}\square C_{5},$$ $$(n\ge 3).$$ Additionally, for $$n=3,4,5,6,$$ we show that $$\gamma _{r2}(C_{n}\square C_{5})=2n.$$

The domination number of Cartesian product of two directed paths

Let ?(P m ?P n ) be the domination number of the Cartesian product of directed paths P m and P n for m,n?2. Liu et al. in (J. Comb. Optim. 22(4):651---662, 2011) determined the value of ?(P m ?P n ) for arbitrary n and m≤6. In this work we give the exact value of ?(P m ?P n ) for any m,n and exhibit dominating sets of minimum cardinality.

Domination number of Cartesian products 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 Liu et al. (2010) [11], we determined the exact values of γ ( C m □ C n ) when m = 2 , 3 , 4 . In this paper, we give a lower and upper bounds for γ ( C m □ C n ) . Furthermore, we prove a necessary and sufficient conditions for C m □ C n to have an efficient dominating set. Also, we determine the exact values: γ ( C 5 □ C n ) = 2 n ; γ ( C 6 □ C n ) = 2 n if n ≡ 0 ( mod 3 ) , otherwise, γ ( C 6 □ C n ) = 2 n + 2 ; γ ( C m □ C n ) = m n 3 if m ≡ 0 ( mod 3 ) and n ≡ 0 ( mod 3 ) .