Abstract
A wide range of parameters of domination in graphs can be defined and studied through a common approach that was recently introduced in [https://doi.org/10.26493/18553974.2318.fb9] under the name of wdomination, where \(w=(w_0,w_1, \dots ,w_l)\) is a vector of nonnegative integers such that \( w_0\ge 1\). Given a graph G, a function \(f: V(G)\longrightarrow \{0,1,\dots ,l\}\) is said to be a wdominating function if \(\sum _{u\in N(v)}f(u)\ge w_i\) for every vertex v with \(f(v)=i\), where N(v) denotes the open neighbourhood of \(v\in V(G)\). The weight of f is defined to be \(\omega (f)=\sum _{v\in V(G)} f(v)\), while the wdomination number of G, denoted by \(\gamma _{w}(G)\), is defined as the minimum weight among all wdominating functions on G. A wide range of wellknown domination parameters can be defined and studied through this approach. For instance, among others, the vector \(w=(1,0)\) corresponds to the case of standard domination, \(w=(2,1)\) corresponds to double domination, \(w=(2,0,0)\) corresponds to Italian domination, \(w=(2,0,1)\) corresponds to quasitotal Italian domination, \(w=(2,1,1)\) corresponds to total Italian domination, \(w=(2,2,2)\) corresponds to total \(\{2\}\)domination, while \(w=(k,k1,\dots ,1,0)\) corresponds to \(\{k\}\)domination. In this paper, we show that several domination parameters of lexicographic product graphs \(G\circ H\) are equal to \(\gamma _{w}(G)\) for some vector \(w\in \{2\}\times \{0,1,2\}^{l}\) and \(l\in \{2,3\}\). The decision on whether the equality holds for a specific vector w will depend on the value of some domination parameters of H. In particular, we focus on quasitotal Italian domination, total Italian domination, 2domination, double domination, total \(\{2\}\)domination, and double total domination of lexicographic product graphs.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The lexicographic product of two graphs G and H is the graph \(G \circ H\) whose vertex set is \(V(G \circ H)= V(G) \times V(H )\) and \((g,h)(g',h') \in E(G \circ H)\) if and only if \(gg' \in E(G)\) or \(g=g'\) and \(hh' \in E(H)\). For simplicity, the neighbourhood of \((x,y)\in V(G)\times V(H)\) will be denoted by N(x, y) instead of N((x, y)). Analogously, for any function f on \(G\circ H\), the image of \((x,y)\in V(G)\times V(H)\) will be denoted by f(x, y) instead of f((x, y)). For basic properties of the lexicographic product of two graphs, we cite the books [18, 23]. In particular, for results on domination theory of lexicographic product graphs we suggest the following works: standard domination [25, 26], Roman domination [27], weak Roman domination [6, 24, 29], total Roman domination [8, 12], total weak Roman domination [6, 11], rainbow domination [28], super domination [14], Italian domination [5], secure domination [6, 24], secure total domination [6, 11], double domination [9] and doubly connected domination [2].
In particular, the next theorem merges two results obtained in [27] and [30]. The result states that the domination number of \(G\circ H\) equals the domination number of G whenever H has domination number equal to one, while the domination number of \(G\circ H\) equals the total domination number of G for the remaining cases.
Theorem 1
([27] and [30]) For any graph G with no isolated vertex and any nontrivial graph H,
Another interesting result obtained in [11] concerns the case of total domination.
Theorem 2
[11] For any graph G with no isolated vertex and any nontrivial graph H,
These two theorems suggest to consider the following problem.
Problem 1
Let G be a graph and let \(\gamma _y\) be a domination parameter well defined on \(G\circ H\) for any nontrivial graph H. Determine if for each graph H, there exists a domination parameter \(\gamma _x\) such that
We proceed to show other cases for which this problem has been solved. To this end, we need to formalize the notion of wdomination introduced in [5], where \(w=(w_0,w_1, \dots ,w_l)\) is a vector of nonnegative integers such that \( w_0\ge 1\). Given a graph G, a function \(f: V(G)\longrightarrow \{0,1,\dots ,l\}\) is said to be a wdominating function if \(\sum _{u\in N(v)}f(u)\ge w_i\) for every vertex v with \(f(v)=i\), where N(v) denotes the open neighbourhood of \(v\in V(G)\). For every \(i\in \{0,\dots , l\}\), we define \(V_i=\{v\in V(G):\; f(v)=i\}\), and we will identify the function f with the subsets \(V_0,\dots ,V_l\) associated with it. So, we will use the unified notation \(f(V_0,\dots , V_l)\) for the function and these associated subsets. The weight of f is defined to be \(\omega (f)=\sum _{v\in V(G)} f(v)\), while the wdomination number of G, denoted by \(\gamma _{_w}(G)\), is defined as the minimum weight among all wdominating functions on G. A wdominating function of weight \(\gamma _{_w}(G)\) will be called a \(\gamma _{_w}(G)\)function.
It was shown in [5] that a wide range of wellknown domination parameters can be defined and studied through this approach. For instance, the vector \(w=(1,0)\) corresponds to standard domination, \(w=(1,1)\) corresponds to total domination, \(w=(2,0,0)\) corresponds to Italian domination, \(w=(2,0,1)\) corresponds to quasitotal Italian domination, \(w=(2,1,1)\) corresponds to total Italian domination, while \(w=(k,k1,\dots ,1,0)\) corresponds to \(\{k\}\)domination.
As the next result shows, Problem 1 was solved for the case of the Italian domination number, which is a wellknown parameter introduced in [13] under the name of Roman \(\{2\}\)domination number. As mentioned above, in terms of wdomination, the Italian domination number of a graph G is defined as \(\gamma _{_I}(G)=\gamma _{_{(2,0,0)}}(G)\).
Theorem 3
[5] For any graph G with no isolated vertex and any nontrivial graph H,
In addition, Problem 1 was solved for the case of the \(\{2\}\)domination number, which was introduced in [15]. In terms of wdomination, the \(\{2\}\)domination number of a graph G is defined as \(\gamma _{_{\{2\}}}(G)=\gamma _{_{(2,1,0)}}(G)\).
Theorem 4
[4] For any graph G with no isolated vertex and any nontrivial graph H,
We refer the reader to [5] for general results on wdomination, as well as for specific results on the domination parameters given in Theorems 3 and 4.
In this paper, we solve Problem 1 for the particular cases in which \(\gamma _y\) corresponds to the following parameters. Although we will use the standard notation for these parameters, we will define them in terms of wdomination.

The kdomination number of a graph G, introduced in [16, 17], can be defined as \(\gamma _{_{k}}(G)=\gamma _{_{(k,0)}}(G)\). In this paper, we are interested in the case \(k=2\), which is probably the most studied. In this case, if \(f(V_0,V_1)\) is a \(\gamma _{(2,0)}(G)\)function, then we will say that \(V_1\) is a \(\gamma _{_2}(G)\)set.

The double domination number of a graph G with no isolated vertex is defined to be \(\gamma _{_{\times 2}}(G)=\gamma _{_{(2,1)}}(G)\). If \(f(V_0,V_1)\) is a \(\gamma _{_{(2,1)}}(G)\)function, then we will say that \(V_1\) is a \(\gamma _{_{\times 2}}(G)\)set. This parameter was introduced in two different papers [19, 20]. Moreover, the general version of this parameter, the ktuple domination number, is defined to be \(\gamma _{_{\times k}}(G)=\gamma _{_{(k,k1)}}(G)\).

The double total domination number of a graph G with minimum degree \(\delta (G)\ge 2\) is defined to be \(\gamma _{_{\times 2,t}}(G)=\gamma _{_{(2,2)}}(G)\). If \(f(V_0,V_1)\) is a \(\gamma _{_{(2,2)}}(G)\)function, then we will say that \(V_1\) is a \(\gamma _{_{\times 2,t}}(G)\)set. This domination parameter was introduced in [21], and its general version is the ktuple total domination number, which is defined to be \(\gamma _{_{\times k, t}}(G)=\gamma _{_{(k,k)}}(G)\).

The quasitotal Italian domination number of a graph G, recently introduced in [7], is defined to be \(\gamma _{_{I^*}}(G)=\gamma _{_{(2,0,1)}}(G)\). A (2, 0, 1)dominating function of weight \(\gamma _{_{I^*}}(G)\) will be called a \(\gamma _{_{I^*}}(G)\)function.

The total Italian domination number of a graph G with no isolated vertex is defined to be \(\gamma _{_{tI}}(G)=\gamma _{_{(2,1,1)}}(G)\). This parameter was introduced in [3], and independently in [1], under the name of total Roman \(\{2\}\)domination number. A (2, 1, 1)dominating function of weight \(\gamma _{_{tI}}(G)\) will be called a \(\gamma _{_{tI}}(G)\)function.

The total \(\{2\}\)domination number of a graph G of minimum degree \(\delta (G)\ge 2\) is defined as \(\gamma _{_{\{2\},t}}(G)=\gamma _{_{(2,2,2)}}(G)\). This parameter was studied in [22].
We will show that the abovementioned domination parameters of lexicographic product graphs \(G\circ H\) are equal to \(\gamma _{w}(G)\) for some vector \(w\in \{2\}\times \{0,1,2\}^{l}\) and \(l\in \{2,3\}\). The decision on whether the equality holds for a specific vector w will depend on the value of some domination parameters of H.
Notice that if G is a graph with no isolated vertex and H is a nontrivial graph, then the following domination chain is deduced by the definition of the parameters involved in it.
Furthermore, the equality \(\gamma _{_{tI}}(G\circ H)=\gamma _{_{\times 2}}(G\circ H)\) was deduced in [9], while the equality \(\gamma _{_{I^*}}(G\circ H)=\gamma _{_{2}}(G\circ H)\) will be proved in Sect. 2 and the equality \(\gamma _{_{\{2\},t}}(G\circ H)=\gamma _{_{\times 2,t}}(G\circ H)\) will be proved in Sect. 4. Therefore, the following domination chain holds whenever G is a graph with no isolated vertex and H is a nontrivial graph.
2 Double Domination and Total Italian Domination
To get our results, we need to set up some tools and introduce some known results.
Lemma 1
Let G be a graph with no isolated vertex and H a nontrivial graph. If \(\gamma _{_{\times 2}}(H)=2\), then \(\gamma _{_{\times 2}}(G\circ H)\le \gamma _{_{(2,1,0)}}(G)\).
Proof
Let \(S=\{v_1,v_2\}\) be a \(\gamma _{_{\times 2}}(H)\)set and \(g(W_0,W_1,W_2)\) a \(\gamma _{_{(2,1,0)}}(G)\)function. Since \(W=(W_1\times \{v_1\})\cup (W_2\times S)\) is a double dominating set of \(G\circ H\), we conclude that \(\gamma _{_{\times 2}}(G\circ H)\le \vert W\vert =\omega (g)=\gamma _{_{(2,1,0)}}(G)\). \(\square \)
Notice that for any \(u\in V(G)\) the subgraph of \(G\circ H\) induced by \(\{u\}\times V(H)\) is isomorphic to H. For simplicity, we will denote this subgraph by \(H_u\).
Theorem 5
[9] The following statements hold for any graph G with no isolated vertex and any nontrivial graph H.

(i)
\(\gamma _{_{\times 2}}(G\circ H)=\gamma _{_{tI}}(G\circ H).\)

(ii)
If \(\gamma _{_2}(H)\ge 3\) and \(\gamma (H)=1\), then \(\gamma _{_{\times 2}}(G\circ H)=\gamma _{_{tI}}(G).\)

(iii)
There exists a \(\gamma _{_{\times 2}}(G\circ H)\)set S such that \(\vert S \cap V(H_u)\vert \le 2\), for every \(u\in V(G)\).
By Theorem 5 (i), we will restrict the proof of the next result to obtain the values of \( \gamma _{_{\times 2}}(G\circ H)\).
Theorem 6
For any graph G with no isolated vertex and any nontrivial graph H,
Proof
First, we assume that \(\gamma (H)=1\). Since \(\gamma _{_{ I}}(G\circ H)\le \gamma _{_{\times 2}}(G\circ H)\), if \(\gamma _{_{\times 2}}(H)=2\), then Theorem 3 and Lemma 1 lead to \(\gamma _{_{(2,1,0)}}(G)= \gamma _{_{I}}(G\circ H)\le \gamma _{_{\times 2}}(G\circ H)\le \gamma _{_{(2,1,0)}}(G)\). Therefore, in this case we conclude that \(\gamma _{_{\times 2}}(G\circ H)=\gamma _{_{(2,1,0)}}(G)\). Now, if \(\gamma _{_{2}}(H)\ge 3\), then Theorem 5 (ii) leads to \( \gamma _{_{\times 2}}(G\circ H)=\gamma _{_{tI}}(G)=\gamma _{_{(2,1,1)}}(G)\).
From now on we assume that \(\gamma (H)\ge 2\). Let S be a \(\gamma _{_{\times 2}}(G\circ H)\)set which satisfies Theorem 5 (iii). Let \(f(X_0,X_1,X_2)\) be the function defined on G by \(X_i=\{x\in V(G): \vert S\cap V(H_x)\vert =i\}\) for every \(i \in \{0,1,2\}\). Notice that \(\gamma _{_{\times 2}}(G\circ H)=\vert S\vert =\omega (f)\). We claim that f is a \(\gamma _{_{(2,2,w)}}(G)\)function, where \(w\in \{1,2\}\). In order to prove this claim and find the exact value of w, we differentiate the following two cases.
Case 1. \(\gamma (H)=2\). Assume that \(x\in X_0\cup X_1\). Since \(\gamma (H)=2\), there exists a vertex \(z\in V(H)\) such that \((x,z)\notin S\) and \(\vert S\cap N(x,z)\cap V(H_x)\vert =0\). Hence, \(\vert S\cap (N(x,z)\setminus V(H_x))\vert \ge 2\), which implies that \(f(N(x))\ge 2\). Now, assume that \(x\in X_2\). In this case, there exists a vertex \(y\in V(H)\) such that \(\vert S\cap N(x,y)\cap V(H_x)\vert \le 1\), and so \(f(N(x))\ge 1\). Therefore, f is a (2, 2, 1)dominating function on G and, as a consequence, \(\gamma _{_{\times 2}}(G\circ H)=\vert S\vert =\omega (f)\ge \gamma _{_{(2,2,1)}}(G)\).
Moreover, let \(h(Y_0,Y_1,Y_2)\) be a \(\gamma _{_{(2,2,1)}}(G)\)function and \(S=\{v_1,v_2\}\) a \(\gamma (H)\)set. Notice that the set \(Y=(Y_1\times \{v_1\})\cup (Y_2\times S)\) is a double dominating set of \(G\circ H\), which implies that \(\gamma _{_{\times 2}}(G\circ H)\le \vert Y\vert =\omega (h)=\gamma _{_{(2,2,1)}}(G)\).
Case 2. \(\gamma (H)\ge 3\). Let \(x\in V(G)\). Since \(\gamma (H)\ge 3\), there exists \(y\in V(H)\) such that \((x,y)\notin S\) and \(\vert S\cap N(x,y)\cap V(H_x)\vert =0\), which implies that \(\vert S\cap (N(x,y)\setminus V(H_x))\vert \ge 2\), and so \(f(N(x))\ge 2\). Therefore, f is a (2, 2, 2)dominating function on G and, as a consequence, \(\gamma _{_{\times 2}}(G\circ H)=\vert S\vert =\omega (f)\ge \gamma _{_{(2,2,2)}}(G)\).
It remains to show that \(\gamma _{_{\times 2}}(G\circ H)\le \gamma _{_{(2,2,2)}}(G)\). To see this we only need to observe that for any \(\gamma _{_{(2,2,2)}}(G)\)function \(g(W_0,W_1,W_2)\) and any pair of vertices \(v_1,v_2\in V(H)\), the set \(W=(W_2\times \{v_1,v_2\})\cup (W_1\times \{v_1\})\) is a double dominating set of \(G\circ H\), which implies that \(\gamma _{_{\times 2}}(G\circ H)\le \vert W\vert = \omega (g)=\gamma _{_{(2,2,2)}}(G)\), as required. \(\square \)
3 Quasitotal Italian Domination and 2Domination
To begin this section, we will introduce some basic tools.
Lemma 2
For any graph G with no isolated vertex and any nontrivial graph H with \(\gamma (H)=1\), there exists a \(\gamma _{_{2}}(G\circ H)\)set D satisfying that \(\vert D\cap V(H_u)\vert \le 2\) for every \(u\in V(G)\).
Proof
Given a \(\gamma _{_{2}}(G\circ H)\)set D, we define the set \(R_D=\{x\in V(G): \, \vert D\cap V(H_x)\vert \ge 3\}\). Now, we assume that D is a \(\gamma _{_{2}}(G\circ H)\)set such that \(\vert R_D\vert \) is minimum among all \(\gamma _{_{2}}(G\circ H)\)sets. Suppose that \(\vert R_D\vert \ge 1\). Let v be a universal vertex of H and \(u\in R_D\). Now, we take \(u'\in N(u)\) and \(v'\in N(v)\), and consider a set \(D'\subseteq V(G)\times V(H)\) satisfying the following properties.

\(D'\cap V(H_u)=\{(u,v),(u,v')\}\);

\(\vert D'\cap V(H_{u'})\vert =\min \{2, \vert D\cap V(H_{u'})\vert +1\}\);

\(D'\cap V(H_x)=D\cap V(H_x)\) for every \(x\in V(G){\setminus } \{u,u'\}\).
Observe that \(D'\) is a 2dominating set of \(G\circ H\) satisfying \(\vert D'\vert \le \vert D\vert \) and \(\vert R_{D'}\vert <\vert R_D\vert \), which is a contradiction. Therefore, \(R_D=\varnothing \), as required. \(\square \)
Lemma 3
Let G be a graph with no isolated vertex and H a nontrivial graph. If \(\gamma _{_{2}}(H)\ge 3\) and \(\gamma (H)=1\), then \(\gamma _{_{2}}(G\circ H)\ge \gamma _{_{(2,1,1)}}(G).\)
Proof
Let D be a \(\gamma _{_{2}}(G\circ H)\)set which satisfies Lemma 2. Let \(f(X_0,X_1,X_2)\) be the function defined on G by \(X_i=\{x\in V(G): \vert D\cap V(H_x)\vert =i\}\) for every \(i\in \{0,1,2\}\). Notice that \(\gamma _{_{2}}(G\circ H)=\vert D\vert =\omega (f)\). We claim that f is a (2, 1, 1)dominating function on G. Assume that \(x\in X_0\). Since \(D\cap V(H_x)=\varnothing \), we have that \(\vert D\cap (N(x)\times V(H)\vert \ge 2\), which implies that \(f(N(x))\ge 2\). Now, assume that \(x\in X_1\cup X_2\). Since \(\vert D\cap V(H_x)\vert \le 2\) and \(\gamma _{_{2}}(H)\ge 3\), there exists \(y\in V(H)\) such that \((x,y)\not \in D\) and \(\vert D\cap V(H_x) \cap N(x,y) \vert \le 1\), which implies that \(\vert D\cap (N(x)\times V(H))\vert \ge 1\), and so \(f(N(x))\ge 1\). Therefore, f is a (2, 1, 1)dominating function on G and, as a consequence, \(\gamma _{_{2}}(G\circ H)=\vert D\vert =\omega (f)\ge \gamma _{_{(2,1,1)}}(G)\). \(\square \)
Theorem 7
The following statements hold for any graph G with no isolated vertex and any nontrivial graph H.

(i)
\(\gamma _{_{I^*}}(G\circ H)=\gamma _{_{2}}(G\circ H).\)

(ii)
If \(\gamma (H)\ge 2\), then \(\gamma _{_{I^*}}(G\circ H)=\gamma _{_{I}}(G\circ H).\)
Proof
By definition, \(\gamma _{_{I^*}}(G\circ H)\le \gamma _{_{2}}(G\circ H)\). Hence, it remains to show that \(\gamma _{_{I^*}}(G\circ H)\ge \gamma _{_{2}}(G\circ H)\). Let \(f(V_0,V_1,V_2)\) be a \(\gamma _{_{I^*}}(G\circ H)\)function such that \(\vert V_2\vert \) is minimum among all \(\gamma _{_{I^*}}(G\circ H)\)functions. If \(V_2=\varnothing \), then \(V_1\) is a 2dominating set of \(G\circ H\), and so \(\gamma _{_{2}}(G\circ H)\le \vert V_1\vert = \gamma _{_{I^*}}(G\circ H)\). We assume that \(V_2\ne \varnothing \) and, in that case, we differentiate the next two cases for a fixed vertex \((u,v)\in V_2\). Obviously, \(N(u,v)\cap (V_1\cup V_2)\ne \varnothing \).
Case 1. \(N(u,v)\cap (V_1\cup V_2)\subseteq V(H_u)\). In this case, for any \((u',v')\in N(u)\times V(H)\) we define the function \(f'(V_0',V_1',V_2')\) where \(V_0'=V_0\setminus \{(u',v')\}\), \(V_1'=V_1\cup \{(u,v),(u',v')\}\) and \(V_2'=V_2\setminus \{(u,v)\}\). Observe that \(\omega (f')=\omega (f)\), every vertex in \(V_2'\) has a neighbour in \(V_1'\cup V_2'\) and every vertex \(w\in V_0'\subseteq V_0\) satisfies that \(f'(N(w))\ge 2\). Hence, \(f'\) is a \(\gamma _{_{I^*}}(G\circ H)\)function and \(\vert V_2'\vert <\vert V_2\vert \), which is a contradiction.
Case 2. \((N(u)\times V(H))\cap (V_1\cup V_2)\ne \varnothing \). If \(V(H_u)\subseteq V_1\cup V_2\), then the function h, defined by \(h(u,v)=1\) and \(h(x,y)=f(x,y)\) whenever \((x,y)\in V(G\circ H)\setminus \{(u,v)\}\), is a quasitotal Italian dominating function on \(G\circ H\) with \(\omega (h)<\omega (f)\), which is a contradiction. Hence, there exists \(v'\in V(H)\) such that \((u,v')\in V_0\). In that case, let \(f'(V_0',V_1',V_2')\) be a function defined by \(V_0'=V_0\setminus \{(u,v')\}\), \(V_1'=V_1\cup \{(u,v),(u,v')\}\) and \(V_2'=V_2{\setminus } \{(u,v)\}\). As in the previous case, \(\omega (f')=\omega (f)\), every vertex in \(V_2'\) has a neighbour in \(V_1'\cup V_2'\) and every vertex \(w\in V_0'\subseteq V_0\) satisfies that \(f'(N(w))\ge 2\). Thus, \(f'\) is a \(\gamma _{_{I^*}}(G\circ H)\)function with \(\vert V_2'\vert <\vert V_2\vert \), which is a contradiction again.
According to the two cases above, we deduce that \(V_2=\varnothing \), which implies that \(\gamma _{_{2}}(G\circ H)\le \gamma _{_{I^*}}(G\circ H).\) Therefore, the proof of (i) is complete.
Finally, we proceed to prove (ii). By definition, \(\gamma _{_{I}}(G\circ H)\le \gamma _{_{I^*}}(G\circ H)\). Thus, it remains to show that \(\gamma _{_{I}}(G\circ H)\ge \gamma _{_{I^*}}(G\circ H)\) whenever \(\gamma (H)\ge 2\). Let \(g(W_0,W_1,W_2)\) be a \(\gamma _{_{I}}(G\circ H)\)function such that \(\vert W_2\vert \) is the minimum among all \(\gamma _{_{I}}(G\circ H)\)functions. Obviously, if \(W_2=\varnothing \) or \(N(u,v)\not \subseteq W_0\) for every \((u,v)\in W_2\), then g is a \(\gamma _{_{I^*}}(G\circ H)\)function and we are done. Suppose to the contrary that there exists a vertex \((u,v)\in W_2\) such that \(N(u,v)\subseteq W_0\). Notice that \(g(V(H_u))\ge 3\), as \(\gamma (H)\ge 2\). Thus, we differentiate the next two cases.
Case 1. \(g(V(H_u))\ge 4\). Let \(u'\in N(u)\) and \(v'\in V(H)\setminus \{v\}\). We define a function \(g'(W_0',W_1',W_2')\) on \(G\circ H\) as \(g'(u,v)=g'(u,v')=g'(u',v)=g'(u',v')=1\), \(g'(V(H_{u}){\setminus } \{(u,v), (u,v')\})=g'(V(H_{u'}){\setminus } \{(u',v),(u',v')\})=0\) and \(g'(x,y)=g(x,y)\) for every \(x\in V(G){\setminus } \{u,u'\}\) and \(y\in V(H)\). Notice that \(g'\) is an Italian dominating function on \(G\circ H\) with \(\omega (g')\le \omega (g)\) and \(\vert W_2'\vert <\vert W_2\vert \), which is a contradiction.
Case 2. \(g(V(H_u))=3\). In this case, since \(\gamma (H)\ge 2\), we deduce that \(\gamma _{_{I}}(H)=3\) and \(\gamma (H)=2\) by the minimality of \(W_2\). Let \(\{v_1,v_2\}\) be a \(\gamma (H)\)set and \(u'\in N(u)\). Consider the function \(g'(W_0',W_1',W_2')\) defined as \(g'(u,v_1)=g'(u,v_2)=1\), \(g'(u,v)=0\) for every \(v\in V(H){\setminus } \{v_1,v_2\}\), \(g'(V(H_{u'}))=1\) and \(g'(x,y)=g(x,y)\) for every \(x\in V(G){\setminus } \{u,u'\}\) and \(y\in V(H)\). Notice that \(g'\) is an Italian dominating function on \(G\circ H\) with \(\omega (g')\le \omega (g)\) and \(\vert W_2'\vert <\vert W_2\vert \), which is a contradiction.
Therefore, either \(W_2=\varnothing \) or every vertex in \(W_2\) has a neighbour in \(W_1\cup W_2\), and so \(\gamma _{_{I^*}}(G\circ H)=\gamma _{_{I}}(G\circ H)\). \(\square \)
According to Theorem 7, we can restrict the proof of the next result to obtain the values of \( \gamma _{_{ 2}}(G\circ H)\).
Theorem 8
For any graph G with no isolated vertex and any nontrivial graph H,
Proof
Since \(\gamma _{_{I}}(G\circ H)\le \gamma _{_{2}}(G\circ H)\), if \(\gamma _{_{\times 2}}(H)=2\), then by Lemma 1 and Theorem 3 we have that \(\gamma _{_{(2,1,0)}}(G)=\gamma _{_{I}}(G\circ H)\le \gamma _{_{2}}(G\circ H)\le \gamma _{_{(2,1,0)}}(G)\). Therefore, in this case we obtain \( \gamma _{_{2}}(G\circ H)= \gamma _{_{(2,1,0)}}(G)\).
Now, since \(\gamma _{_{2}}(G\circ H) \le \gamma _{_{\times 2}}(G\circ H)\), if \(\gamma _{_{2}}(H)\ge 3\) and \(\gamma (H)=1\), then Lemma 3 and Theorem 5 (ii) lead to \(\gamma _{_{(2,1,1)}}(G)\le \gamma _{_{2}}(G\circ H) \le \gamma _{_{\times 2}}(G\circ H)=\gamma _{_{(2,1,1)}}(G).\) Therefore, \(\gamma _{_{2}}(G\circ H)=\gamma _{_{(2,1,1)}}(G)\).
Finally, if \(\gamma (H)\ge 2\), then Theorem 7 leads to \(\gamma _{_{2}}(G\circ H)=\gamma _{_{I^*}}(G\circ H)=\gamma _{_{I}}(G\circ H)\) and so we complete the proof by Theorem 3. \(\square \)
4 Double Total Domination and Total \(\{2\}\)Domination
Although in general, \(\gamma _{_{\{2\},t}}(G)\le \gamma _{_{\times 2,t}}(G)\), we show below that for the case of lexicographic product graphs these parameters always coincide.
Theorem 9
For any graph G with no isolated vertex and any nontrivial graph H,
Proof
By definition, \(\gamma _{_{\{2\},t}}(G\circ H)\le \gamma _{_{\times 2,t}}(G\circ H).\) Hence, it remains to show that \(\gamma _{_{\{2\},t}}(G\circ H)\ge \gamma _{_{\times 2,t}}(G\circ H).\) Let \(f(V_0,V_1,V_2)\) be a \(\gamma _{_{\{2\},t}}(G\circ H)\)function such that \(\vert V_2\vert \) is minimum among all \(\gamma _{_{\{2\},t}}(G\circ H)\)functions. If \(V_2=\varnothing \), then \(V_1\) is a double total dominating set of \(G\circ H\), and so \(\gamma _{_{\times 2,t}}(G\circ H)\le \vert V_1\vert = \gamma _{_{\{2\},t}}(G\circ H)\), as required. We assume that \(V_2\ne \varnothing \) and, in that case, we differentiate the next two cases for a fixed vertex \((u,v)\in V_2\). Obviously, \(N(u,v)\cap (V_1\cup V_2)\ne \varnothing \).
Case 1. \(N(u,v)\cap (V_1\cup V_2)\subseteq V(H_u)\). In this case, for any \((u',v),(u',v')\in N(u)\times V(H)\) we define the function \(f'(V_0',V_1',V_2')\) where \(V_0'=V_0\setminus \{(u',v),(u',v')\}\), \(V_1'=V_1\cup \{(u',v),(u',v')\}\) and \(V_2'=V_2{\setminus } \{(u,v)\}\). Observe that \(\omega (f')=\omega (f)\) and every vertex \((x,y)\in V(G\circ H)\) satisfies that \(f'(N(x,y))\ge 2\). Hence, \(f'\) is a \(\gamma _{_{\{2\},t}}(G\circ H)\)function and \(\vert V_2'\vert <\vert V_2\vert \), which is a contradiction.
Case 2. \((N(u)\times V(H))\cap (V_1\cup V_2)\ne \varnothing \). If \(V(H_u)\subseteq V_1\cup V_2\), then the function h, defined by \(h(u,v)=1\) and \(h(x,y)=f(x,y)\) whenever \((x,y)\in V(G\circ H)\setminus \{(u,v)\}\), is a double total dominating function on \(G\circ H\) with \(\omega (h)<\omega (f)\), which is a contradiction. Hence, there exists \(v'\in V(H)\) such that \((u,v')\in V_0\). In that case, let \(f'(V_0',V_1',V_2')\) be a function defined by \(V_0'=V_0\setminus \{(u,v')\}\), \(V_1'=V_1\cup \{(u,v),(u,v')\}\) and \(V_2'=V_2{\setminus } \{(u,v)\}\). Notice that \(\omega (f')=\omega (f)\) and every vertex \((x,y)\in V(G\circ H)\) satisfies that \(f'(N(x,y))\ge 2\). Thus, \(f'\) is a \(\gamma _{_{\{2\},t}}(G\circ H)\)function with \(\vert V_2'\vert <\vert V_2\vert \), which is a contradiction again.
According to the two cases above, we deduce that \(V_2=\varnothing \), which implies that \(V_1\) is a double total dominating set of \(G\circ H\), and so \(\gamma _{_{\times 2,t}}(G\circ H)\le \vert V_1\vert = \gamma _{_{\{2\},t}}(G\circ H)\), as required. Therefore, the proof is complete. \(\square \)
We are now in a position to formalize the tools which will allow us to calculate \(\gamma _{_{\times 2,t}}(G\circ H)\).
Lemma 4
For any graph G with no isolated vertex and any nontrivial graph H, there exists a \(\gamma _{_{\times 2,t}}(G\circ H)\)set S satisfying that \(\vert S\cap V(H_x)\vert \le 2\) for every \(x\in V(G)\).
Proof
Given a \(\gamma _{_{\times 2,t}}(G\circ H)\)set S, we define the set \(R_S=\{x\in V(G): \, \vert S\cap V(H_x)\vert \ge 3\}\). Assume that S is a \(\gamma _{_{\times 2,t}}(G\circ H)\)set such that \(R_S\) has minimum cardinality among all \(\gamma _{_{\times 2,t}}(G\circ H)\)sets. Suppose that \(R_S\ne \varnothing \) and let \(x,y\in V(G)\) be two adjacent vertices with \(x\in R_S\). Let \(S_x=S\cap V(H_x)\) and take \((x,v_1),(x,v_2)\in S_x\). Hence, there exists a set \(S'\subseteq V(G\circ H)\) satisfying the following properties.

\(S'\cap V(H_x)=\{(x,v_1),(x,v_2)\}\).

\(\vert S'\cap V(H_{y})\vert =\min \{2, \vert S\cap V(H_{y})\vert +\vert S_x\vert 2\}\).

\(S'\cap V(H_{z}) = S\cap V(H_{z}) \) for every \(z\in V(G){\setminus } \{x,y\}\).
Observe that \(S'\) is a double total dominating set of \(G\circ H\) with \(\vert S'\vert \le \vert S\vert \) and \(\vert R_{S'}\vert <\vert R_{S}\vert \), which is a contradiction. Therefore, the result follows. \(\square \)
Proposition 1
For any graph G with no isolated vertex and any nontrivial graph H,
Furthermore, if H has isolated vertex or \(\gamma _{_{t}}(H)\ge 3\), then the equality holds.
Proof
The proof of the inequality is straightforward, as we only need to observe that for any \(\gamma _{_{(2,2,2)}}(G)\)function \(g(W_0,W_1,W_2)\) and any pair of vertices \(v_1,v_2\in V(H)\), the set \(W=(W_2\times \{v_1,v_2\})\cup (W_1\times \{v_1\})\) is a double total dominating set of \(G\circ H\), which implies that \(\gamma _{_{\times 2,t}}(G\circ H)\le \vert W\vert = \omega (g)=\gamma _{_{(2,2,2)}}(G)\).
From now on, assume that either H has isolated vertex or \(\gamma _{_{t}}(H)\ge 3\). Notice that these assumptions imply that for any set \(S\subseteq V(H)\) of cardinality at most two, there exists a vertex \(v\in V(H)\) such that \(N(v)\cap S=\varnothing \).
Now, let D be a \(\gamma _{_{\times 2,t}}(G\circ H)\)set satisfying Lemma 4. Since \(\vert D\cap V(H_x)\vert \le 2\) for every \(x\in V(G)\), from the assumptions above we have that there exists a vertex \(v\in V(H)\) such that \(N(x,v)\cap D \cap V(H_x)=\varnothing \). Thus, \(\vert (N(x)\times V(H))\cap D\vert \ge 2\) for every \(x\in V(G)\), which implies that any function \(f: V(G)\longrightarrow \{0,1,2\}\) such that \(f(V(H_x))=\vert D\cap V(H_x)\vert \), is a (2, 2, 2)dominating function on G. Therefore, \(\gamma _{_{(2,2,2)}}(G)\le \omega (f)=\vert D\vert = \gamma _{_{\times 2,t}}(G\circ H)\), as required. \(\square \)
According to Theorem 9, in the proof of the following result we can restrict ourselves to determining the value of \(\gamma _{_{\times 2,t}}(G\circ H)\).
Theorem 10
For any graph G with no isolated vertex and any nontrivial graph H,
Proof
First we assume that \(\gamma _{_{t}}(H)=2\). Let \(h(Y_0,Y_1,Y_2)\) be a \(\gamma _{_{(2,2,1)}}(G)\)function and let \(S=\{v_1,v_2\}\) be a \(\gamma _{_{t}}(H)\)set. Notice that the set \(Y=(Y_1\times \{v_1\})\cup (Y_2\times S)\) is a double total dominating set of \(G\circ H\), which implies that \(\gamma _{_{\times 2,t}}(G\circ H)\le \vert Y\vert =\omega (h)=\gamma _{_{(2,2,1)}}(G)\). Now, let S be a \(\gamma _{_{\times 2,t}}(G\circ H)\)set which satisfies Lemma 4 and let \(f(X_0,X_1,X_2)\) be the function defined on G by \(X_i=\{x\in V(G): \vert S\cap V(H_x)\vert =i\}\) for every \(i\in \{0,1,2\}\). Notice that \(\gamma _{_{\times 2,t}}(G\circ H)=\vert S\vert =\omega (f)\). We claim that f is a (2, 2, 1)dominating function on G.
Let \(x\in X_0\cup X_1\). Since \(\gamma _{_{t}}(H)=2\), there exists a vertex \(z\in V(H)\) such that \((x,z)\notin S\) and \(\vert S\cap N(x,z)\cap V(H_x)\vert =0\). Hence, as S is a \(\gamma _{_{\times 2,t}}(G\circ H)\)set, \(\vert S\cap (N(x,z){\setminus } V(H_x))\vert \ge 2\), and so \(f(N(x))\ge 2\).
Now, let \(x\in X_2\). Since \(\gamma _{_{t}}(H)=2\) implies \(\gamma _{_{\times 2,t}}(H)\ge 3\), we have that there exists a vertex \(y\in V(H)\) such that \(\vert S\cap V(H_x)\cap N(x,y)\vert \le 1\), which leads to \(\vert S\cap (N(x,z)\setminus V(H_x))\vert \ge 1\), as S is a \(\gamma _{_{\times 2,t}}(G\circ H)\)set, and so \(f(N(x))\ge 1\).
Therefore, f is a (2, 2, 1)dominating function on G and, as a consequence, \(\gamma _{_{\times 2,t}}(G\circ H)=\vert S\vert =\omega (f)\ge \gamma _{_{(2,2,1)}}(G)\), concluding that \(\gamma _{_{\times 2,t}}(G\circ H)=\gamma _{_{(2,2,1)}}(G)\).
Finally, if \(\gamma _{_{t}}(H)\ge 3\) or H has isolated vertex, then by Proposition 1 we have \(\gamma _{_{\times 2,t}}(G\circ H)= \gamma _{_{(2,2,2)}}(G).\) \(\square \)
References
Abdollahzadeh Ahangar, H., Chellali, M., Sheikholeslami, S.M., ValenzuelaTripodoro, J.C.: Total Roman \(\{2\}\)dominating functions in graphs. Discuss. Math. Graph Theory 42(3), 937–958 (2022)
Arriola, B.H., Canoy, S.R., Jr.: Doubly connected domination in the corona and lexicographic product of graphs. Appl. Math. Sci. 8(29–32), 1521–1533 (2014)
Cabrera García, S., Cabrera Martínez, A., Hernández Mira, F.A., Yero, I.G.: Total Roman \(\{2\}\)domination in graphs. Quaest. Math. 44(3), 411–434 (2021)
Cabrera Martínez, A., Conchado Peiró, A.: On the \(\{2\}\)domination number of graphs. AIMS Math. 7(6), 10731–10743 (2022)
Cabrera Martínez, A., EstradaMoreno, A., RodríguezVelázquez, J.A.: From Italian domination in lexicographic product graphs to \(w\)domination in graphs. Ars Math. Contemp. 22(1), 1.0425 (2020)
Cabrera Martínez, A., EstradaMoreno, A., RodríguezVelázquez, J.A.: From (secure) \(w\)domination in graphs to protection of lexicographic product graphs. Bull. Malays. Math. Sci. Soc. 44(6), 3747–3765 (2021)
Cabrera Martínez, A., EstradaMoreno, A., RodríguezVelázquez, J.A.: From the quasitotal strong differential to quasitotal Italian domination in graphs. Symmetry 13(6), 1036 (2021)
Cabrera Martínez, A., RodríguezVelázquez, J.A.: Closed formulas for the total Roman domination number of lexicographic product graphs. Ars MathArs. Math. Contemp. 20, 233–241 (2021)
Cabrera Martínez, A., Cabrera García, S., RodríguezVelázquez, J.A.: Double domination in lexicographic product graphs. Discrete Appl. Math. 284, 290–300 (2020)
Cabrera Martínez, A., Montejano, L.P., RodríguezVelázquez, J.A.: Total weak Roman domination in graphs. Symmetry 11(6), 831 (2019)
Cabrera Martínez, A., RodríguezVelázquez, J.A.: Total protection of lexicographic product graphs. Discuss. Math. Graph Theory 42(3), 967–984 (2022)
Campanelli, N., Kuziak, D.: Total Roman domination in the lexicographic product of graphs. Discrete Appl. Math. 263, 88–95 (2019)
Chellali, M., Haynes, T.W., Hedetniemi, S.T., McRae, A.A.: Roman \(\{2\}\)domination. Discrete Appl. Math. 204, 22–28 (2016)
Dettlaff, M., Lemańska, M., RodríguezVelázquez, J.A., Zuazua, R.: On the super domination number of lexicographic product graphs. Discrete Appl. Math. 263, 118–129 (2019)
Domke, G. S., Hedetniemi, S. T., Laskar, R. C., Fricke, G. H. Relationships between integer and fractional parameters of graphs. In: Yousef Alavi, G. Chartrand, Oellermann, O. R., Schwenk, A. J. (eds.), Graph theory, combinatorics, and applications: In: Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, Western Michigan University, (2) pp. 371–387. John Wiley & Sons Inc., 1991
J. Frederick Fink, M. S. Jacobson, \(n\)domination in graphs, In: Graph theory with applications to algorithms and computer science, WileyIntersci. Publ., 283–300, Wiley, New York, 1985
Fink, J.F., Jacobson, M.S., On \(n\)domination, \(n\)dependence and forbidden subgraphs, In: Graph theory with applications to algorithms and computer science (Kalamazoo, Mich., 1984), WileyIntersci. Publ., 301–311. Wiley, New York, 1985
Hammack, R., Imrich, W., Klavžar, S.: Handbook of product graphs, 2nd edn. Discrete Mathematics and its Applications, CRC Press (2011)
Harary, F., Haynes, T.W.: NordhausGaddum inequalities for domination in graphs. Discrete Math. 155(1–3), 99–105 (1996)
Harary, F., Haynes, T.W.: Double domination in graphs. Ars Combin. 55, 201–213 (2000)
Henning, M.A., Kazemi, A.P.: \(k\)tuple total domination in graphs. Discrete Appl. Math. 158(9), 1006–1011 (2010)
Hou, X., Lu, Y.: On the \(\{k\}\)domination number of cartesian products of graphs. Discrete Math. 309(10), 3413–3419 (2009)
Imrich, W., Klavžar, S., Product graphs, structure and recognition, WileyInterscience series in discrete mathematics and optimization, Wiley, 2000
Klein, D.J., RodríguezVelázquez, J.A.: Protection of lexicographic product graphs. Discuss. Math. Graph Theory 42, 139–158 (2022)
Liu, J., Zhang, X., Meng, J.: Domination in lexicographic product digraphs. Ars Combin. 120, 23–32 (2015)
Nowakowski, R.J., Rall, D.F.: Associative graph products and their independence, domination and coloring numbers. Discuss. Math. Graph Theory 16, 53–79 (1996)
Šumenjak, T.K., Pavlič, P., Tepeh, A.: On the Roman domination in the lexicographic product of graphs. Discrete Appl. Math. 160(13–14), 2030–2036 (2012)
Šumenjak, T.K., Rall, D.F., Tepeh, A.: Rainbow domination in the lexicographic product of graphs. Discrete Appl. Math. 161(13–14), 2133–2141 (2013)
Valveny, M., PérezRosés, H., RodríguezVelázquez, J.A.: On the weak Roman domination number of lexicographic product graphs. Discrete Appl. Math. 263, 257–270 (2019)
Zhang, X., Liu, J., Meng, J.: Domination in lexicographic product graphs. Ars Combin. 101, 251–256 (2011)
Funding
Open access funding provided by Universitat Rovira i Virgili.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Shamani Supramaniam.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
CabreraMartínez, A., Montejano, L.P. & RodríguezVelázquez, J.A. From wDomination in Graphs to Domination Parameters in Lexicographic Product Graphs. Bull. Malays. Math. Sci. Soc. 46, 109 (2023). https://doi.org/10.1007/s40840023015025
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40840023015025
Keywords
 wdomination
 (Total) Italian domination
 Quasitotal Italian domination
 2domination
 Double domination
 Lexicographic product graph