Bounds on the domination number and the metric dimension of co-normal product of graphs

In this paper, we establish bounds on the domination number and the metric dimension of the co-normal product graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G_{H}$\end{document}GH of two simple graphs G and H in terms of parameters associated with G and H. We also give conditions on the graphs G and H for which the domination number of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G_{H}$\end{document}GH is 1, 2, and the domination number of G. Moreover, we give formulas for the metric dimension of the co-normal product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G_{H}$\end{document}GH of some families of graphs G and H as a function of associated parameters of G and H.


Introduction
The domination number is a parameter that has appeared in numerous location problems [19] and in the analysis of social network problems [4]. The adjacency and non-adjacency relation between two vertices u, v in a graph G is denoted by u ∼ v and u v, respectively. A set D ⊆ V (G), is a dominating set [22] of G if for every v ∈ V (G), we have v ∈ D or v ∼ u for some u ∈ D. The minimum cardinality of a dominating set in a graph G is called the domination number of G, denoted by γ (G). The problem of finding a minimum size dominating set of a graph is in general NP-hard [13].
The metric dimension is a parameter that has appeared in robot navigation problems [20], strategies for the mastermind game [8], drug discovery problems [7,17,18], coin weighing problems [26], network discovery and verification problems [3]. The notation d G (u, v) or simply d (u, v) denotes the distance between two vertices u, v ∈ V (G), which is the length of a shortest path between them. For an ordered set W = {w 1 , w 2 , . . . , w k } ⊆ V (G) and a vertex v ∈ V (G), the k-vector (d(v, w 1 ), d(v, w 2 ), . . . , d(v, w k )), is called the metric representation of v with respect to W , denoted by c W (v). A set W ⊆ V (G) is a resolving (locating) set [14,27] of G if for any two distinct vertices u, v ∈ V (G), c W (u) = c W (v), which means that there exists at least one vertex w ∈ W for which d(v, w) = d (u, w). A minimum resolving set of G is called a metric basis of G and its cardinality is called the metric dimension of G, denoted by dim(G)(loc(G)). Gary and Johnson [13] noted that the problem of finding the metric dimension of a graph is NP-hard; however, its explicit construction is given by Khuller et al. [20]. The problem of finding the metric dimension of a graph is formulated as an integer programming problem by Chartrand et al. [7]. Relations between the domination number and the metric dimension of a graph are given in [1].
It is found in [2] that there are 256 possible products of any two graphs using the adjacency and the non-adjacency relations of these graphs. Several interesting types of graph products have been studied extensively in the literature. For instance, Caceres et al. [6], Yero et al. [29], Rodriguez-Velazquez et al. [24], Saputroa et al. [25], and Jannesari and Omoomi [16] investigated the metric dimension of the cartesian product, the corona product, the strong product, and the lexicographic product of graphs, respectively.
Out of product graphs, there is another well-known product graph introduced by Ore in 1962 [22], with the name cartesian sum of graphs. It was named co-normal product of graphs in [12]. Different properties and results regarding coloring and the chromatic number of the co-normal product of graphs are discussed in [5,9,11,12,23,28]. In [21], Kuziak et al. studied the strong metric dimension of the co-normal product of graphs using the strong metric dimension of its components. In this paper, we have studied the domination number and the metric dimension of the co-normal product of graphs.
All considered graphs in this paper are non-trivial, simple and finite. In the next section, we describe some structural properties of the co-normal product of graphs. In Sect. 3, we study the domination number of the co-normal product of graphs and describe conditions on the graphs G and H so that the domination number of G H is 1, 2, and γ (G). We also give bounds on the domination number of the co-normal product of graphs. In Sect. 4, we describe some properties of resolving sets in the co-normal product of graphs and give bounds on the metric dimension of the co-normal product of graphs. Moreover, we establish formulas for the metric dimension of some families of graphs.

Methods
We use the combinatorial computing, combinatorial inequalities and graph theoretic analytic methods to prove the main results. The aim of this research is to provide bounds on the domination number and the metric dimension of the co-normal product of graphs and to give exact formulas for the metric dimension of some families of graphs.

Co-normal product of graphs
The co-normal product (the terminology we have adopted) of a graph G of order m with the vertex set V (G) = {v 1 , v 2 , . . . , v m } and a graph H of order n with the vertex set All results given in this paper for G H also hold for H G due to the commutativity of this product. Figure 1 shows the co-normal product graph G H of two path graphs.
A graph having n vertices in which each vertex is adjacent to all other vertices is called a complete graph, denoted by K n . In [12], Frelih and Miklavic discussed the connectivity of G H and proved the following theorem.

Theorem 1 (Frelih and Miklavic) G H is connected if and only if one of the following holds:
(1) H = K n for some n ≥ 2 and G is connected.
(2) G = K m for some m ≥ 2 and H is connected.
(3) G and H are not null graphs and at least one of G or H is without isolated vertices.

Figure 1
The co-normal product graph of P 4 and P 4 The diameter of a graph G, denoted by diam(G), is the maximum distance between any two vertices of G. If G is a disconnected graph then diam(G H ) = ∞. A graph having n vertices and no edges is called a null graph, denoted by N n . In [21], Kuziak et al. discussed the diameter of G H and proved the following theorem.
Theorem 2 (Kuziak, Yero, Rodriguez-Velazquez) Let G and H be two non-trivial graphs such that at least one of them is non-complete and let n ≥ 2 be an integer. Then the following assertions hold: (1) diam(G N n ) = max{2, diam(G)}.
In the next two observations, we give formulas for the degree and the neighborhood of a vertex in G H using the structure of the co-normal product of graphs.

Observation 2 For any vertex
Two vertices having the same neighbors are called false twins. In the next theorem, we describe conditions for any two distinct vertices in G H to be false twins.

Theorem 3 For any two distinct vertices v ij and v rs in G H
The converse follows from the definition of the co-normal product of graphs.
is an equivalence class of false twins in G H . Using Observation 2, we have the following straightforward lemma.

Lemma 1 For any vertex v ij
are equivalence classes of false twins in G and H, respectively.

Domination in co-normal product of graphs
A vertex of a graph G is a dominating vertex if its degree is |V (G)| -1. Throughout this section and the next section, the graphs G, H and G H are as described in Sect. 2. We define vertex sets, Fig. 1, we represent such classes. In the next two results, we give conditions on G and H for which G H have domination numbers 1 or 2.

Lemma 2 A vertex v ij is a dominating vertex in G H if and only if v i and u j are dominating vertices in G and H, respectively.
Proof Let v ij be a dominating vertex in G H . To show that v i , u j are dominating in G and H, respectively, assume contrary that Now suppose that v i and u j are dominating vertices in G and H, respectively, then, by Observation 1, we have deg(v ij ) = |V (G)| · |V (H)| -1.

Lemma 3 If G has a dominating vertex and H has no dominating vertex, then
The total domination number, denoted by γ t (G), is the cardinality of a minimum total dominating set for G. In the next theorem, we give conditions on G and H so that γ (G H ) = γ (G), by using the total domination number of G.

Theorem 4 For any two connected graphs G and H with
Hence, D is dominating set for G H . Now to prove that D is a minimum dominating set, assume contrarily that Lemma 2, shows that γ (G H ) = 1 if and only if γ (G) = γ (H) = 1. In the next theorem, we give general bounds on the domination number of G H .

Theorem 5 For any two connected graphs G and H
. . ,ú n 2 } be dominating sets for G, H, respectively and D = D 1 × D 2 . To show that D is a dominating set for G H , consider a vertex v ij ∈ V (G H ), we have following cases: Note that the lower bound given in Theorem 5, is attainable when γ (G) = γ (H).

Metric dimension in co-normal product of graphs
In this section, we study the properties of resolving sets in G H and establish formulas for the co-normal product of some families of graphs. In Theorem 10, we give bounds on the metric dimension of the co-normal product of a connected graph G and a graph H (not necessarily connected). In the rest of this paper, we assume G and H such that G H is connected. Moreover, G H has diameter at most two unless otherwise stated. In the next lemma, we will prove that, for every v i ∈ V (G), u j ∈ V (H) and an ordered set

Lemma 4 Let G H has diameter 2 and W
; v l ∈ V (G)} gives a partition of W . For any vertex v ij ∈ V (G H ), the code of v ij with respect to W can be represented as: In the next lemma, we give conditions on an ordered set W ⊆ V (G H ) to be a resolving set for G H .

Lemma 5 A set W ⊆ V (G H ) is a resolving set for G H if and only if for any two distinct vertices v ij
Proof Suppose W is a resolving set for G H and there exist two distinct vertices v ij , v rs in G H such that, for every v l in G, Conversely, suppose for any two distinct vertices v ij , v rs ∈ V (G H ), there exists at least one showing that W is a resolving set for G H .
In [15], the authors proved the following corollary, which gives the relation between resolving sets and false twins of a graph.
In the next theorem, we give conditions on G and H for which there exists a resolving set W of G H such that W ∩ H(v i ) = ∅ for some v i ∈ V (G).

Theorem 6 Let G be a connected graph and H be an arbitrary graph such that diam(G), diam(H) ≥ 2. There exists a resolving set W for G H such that W ∩ H(v i ) = ∅ for some v i ∈ V (G) if and only if H has no false twins.
Proof Let W be a resolving set of G H such that W ∩ H(v i ) = ∅, for some v i ∈ V (G). Assume contrary that N(u j ) = N(u s ) for two distinct vertices u j , u s ∈ V (H), then, by Lemma 3, where v i is not a dominating vertex in G. To prove the converse, we only need to prove that W is a resolving set for G H . Let v ij , v il ∈ H(v i ) be two distinct vertices for some u j , u l ∈ V (H). Since H have no false twins and diameter at least 2, there exists at least one vertex, say u r ∈ V (H), such that u r ∈ N(u j ) or u r ∈ N(u l ). Now for every v k v i in G, we have v kr ∈ N(v ij ) or v kr ∈ N(v il ), which shows that c W (v ij ) = c W (v il ). Hence, W is a resolving set for G H .
The following corollary directly follows from Theorem 6, which gives the relation between dominating sets and resolving sets of G H , when both G, H are connected.

Corollary 2 For any two connected graphs G and H if at least one of G, H has false twins, then every resolving set of G H is a dominating set of G H .
In the next theorem, we give conditions on G and H for which the metric dimension of G H is the order of G times the metric dimension of H. Proof Since N(v i ) = N(v k ), for any two distinct vertices v i , v k ∈ V (G), G has |V (G)| distinct equivalence classes of false twins. Lemma 1, shows that the co-normal product G H has |V (G)| · k equivalence classes of false twins such that no class has cardinality 1, so Let P m ; m ≥ 4 be a path graph and K n 1 ,n 2 ,...,n k ; n i ≥ 2 for each i, be a complete multipartite graph have k distinct equivalence classes of false twins. Since P m have no false twins, by Theorem 7, we have the following corollary.
In [16], Jannesari and Omoomi introduced the concept of the adjacency metric dimension of a graph and used it to find the metric dimension of lexicographic product of graphs. A function a : V (G) × V (G) → {0, 1, 2} defined as: for u, v ∈ V (G), is called the adjacency function of G. The k-vector (a(v, w 1 ), a(v, w 2 ), . . . , a(v, w k )) for a vertex v ∈ V (G), is called the adjacency metric representation of v with respect to W , denoted by c a W (v). A set W is an adjacency resolving set for G if for any two A minimum adjacency resolving set of G is called an adjacency basis of G and its cardinality is called the adjacency metric dimension of G, denoted by adim(G). They also gave that if G is a connected graph with diameter 2, then dim(G) = adim(G) but the converse is not true because dim(C 6 ) = 2 = adim(C 6 ), while diam(C 6 ) = 3. Our next lemma directly follows from the definition of adjacency basis and the fact that the induced subgraph H(v i ) of G H is isomorphic to H, for each v i ∈ V (G).

Lemma 6 If G H has diameter at most 3 and W 2 is an adjacency basis for H, then, for any
Now consider a path graph P 4 having the vertex set V (P 4 ) = {v 1 , v 2 , v 3 , v 4 } such that v i ∼ v i+1 ; i ≤ 3 and a star graph S 4 having the vertex set V (S 4 ) = {u 1 , u 2 , u 3 , u 4 , u 5 } such that u 5 ∼ u i ; 1 ≤ i ≤ 4. The co-normal product graph of P 4 and S 4 is shown in Fig. 2. Note that, for every adjacency basis W 2 of S 4 , c a W 2 (u 5 ) = (1, 1, 1) and is not a resolving set for G H . Let 1 represents a vector whose each entry is 1 and 2 represents a vector whose each entry is 2, i.e. 1 = (1, 1, . . . , 1) and 2 = (2, 2, . . . , 2). In the next theorem, we provide conditions under which W = v i ∈V (G) W 2 (v i ) is a resolving set for G H , where W 2 is an adjacency basis of H and W 2 (v i ) = {v i } × W 2 .

Theorem 8 Let G be a connected graph having no false twins and H be a graph such that G H has diameter at most three. If there exists an adjacency basis W 2 of H such that c a
W 2 (u j ) = 1 for all u j ∈ V (H), then dim(G H ) ≤ |V (G)| · adim(H).
By Lemma 6, W 2 (v i ) resolves all the vertices of H(v i ). To show that W is a resolving set for G H , consider two distinct vertices v ij , v kl ∈ V (G H ) \ W such that v i = v k . Since G has no false twins, we In the next theorem, we give a formula for the metric dimension of G H when G is complete and H is a graph for which each adjacency basis W 2 has one vertex u j ∈ V (H) \ W 2 such that c a W 2 (u j ) = 1.

Theorem 9
Let G be a complete graph and H = K n be an arbitrary graph. If for each adjacency basis W 2 of H, there exists a vertex u j ∈ V (H) \ W 2 such that c a W 2 (u j ) = 1, then dim(G H ) = |V (G)| · (adim(H) + 1) -1.
Proof By using Lemma 6, In the next theorem, we give bounds for the metric dimension of G H when G and H are non-trivial and at least one is not a complete graph.
Theorem 10 Let G be a connected graph and H = K n be an arbitrary graph, then To prove that W is a resolving set for G H , we discuss the following cases: Case 2: Let u j = u l and W 1 ( Case 3: Let v i = v k and u j = u l . Since W 1 and W 2 are adjacency bases for G and H, As W 2 is an adjacency basis for H so there exists at least one vertex u j ∈ W 2 such that c W 2 \{u j } (u j ) = c W 2 \{u j } (u l ) for some u l ∈ V (H) \ W 2 . Also, W ∩ H(v i ) = ∅ for v i ∈ V (G) \ W 1 and the definition of the co-normal product graph gives c W (v ij ) = c W (v il ). Hence, W is not a resolving set for G H .
For a complete graph G and a null graph H, Theorem 2(1) shows that diam(G H ) = 2 and the metric dimension of G H is given in the next theorem.

Theorem 11 If G is a complete graph and H is a null graph, then
Proof Let V (G) = {v 1 , v 2 , . . . , v m } and V (H) = {u 1 , u 2 , . . . , u n }. It is clear from the definition of co-normal product that, for each v i , N(v ij ) = N(v ik ) for all 1 ≤ j, k ≤ n. So any resolving set must contain at least n -1 vertices from each H(v i ), which shows that dim(G H ) ≥ m(n -1). Since H is a null graph, we have c H(v i )\{v ij } (v ij ) = 2 for each i and c H(v k ) (v ij ) = 1 for each k = i, which shows that any subset of V (G H ) containing n -1 vertices from each H(v i ) will be a resolving set for G H . Hence, dim(G H ) = m(n -1).
In the next theorem, we give formula for the metric dimension of G H when G is a path graph and H is a star graph.

Theorem 12
For any two integers m, n ≥ 2, if G is a path graph and H is a star graph having order m and n + 1 respectively, then dim(G H ) = m · dim(H) + adim(G).
Proof Let V (G) = {v 1 , v 2 , . . . , v m } and V (H) = {u 0 , u 1 , u 2 , . . . , u n }, where deg(u 0 ) = n in H. Also, N(u k ) = N(u l ) for all 1 ≤ k, l ≤ n, by using Lemma 3, we have N(v ik ) = N(v il ) for each i. So, any resolving set W for G must contain at least n -1 vertices from each H(v i ). Since deg(u 0 ) = n, by the definition of a co-normal product d(v i0 , v ij ) = 1 for all 1 ≤ i ≤ m and 1 ≤ j ≤ n, which means that the vertices of G(u 0 ) are not resolved by any of v ij , 1 ≤ i ≤ m, 1 ≤ j ≤ n. Also, d(v i0 , v j0 ) ≤ 2 in G H and induced subgraph of G(u 0 ) is isomorphic to G so we must choose adim(G) vertices from G(u 0 ), which shows that dim(G H ) = m · dim(H) + adim(G).

Conclusions
To study the product graphs with respect to graph theoretic parameters is always an important problem. In this paper, we have studied two parameters, the domination number and the metric dimension of the co-normal product of two graphs G and H. These two parameters have a lot of applications in networks and facility location problems. We have given conditions on G and H under which the graph G H has the domination number 1, 2 and γ (G). We also proved that, for any two connected graphs G and H, min{γ (G), γ (H)} ≤ γ (G H ) ≤ γ (G)γ (H). We described some properties of resolving sets of G H and gave conditions on G and H such that dim(G H ) = |V (G)| · dim(H). We have also given conditions on G and H under which dim(G H ) ≤ |V (G)| · adim(H). For a complete graph G and a non-complete graph H, we have given conditions on H under which dim(G H ) = |V (G)| · adim(H) and dim(G H ) = |V (G)| · (adim(H) + 1) -1. For a connected graph G and a non-complete graph H, we proved that adim(H) · adim(G) < dim(G H ) ≤ |V (G)| · adim(H) + |V (H)| · adim(G). We have also given explicit formulas for the metric dimension of the co-normal product of a path graph and a complete multipartite graph, a complete graph and a null graph, a path graph and a star graph for the first time. Our derived inequality relations can be very helpful in the characterizations of graphs with given metric dimension or given domination number.