RESOLVING, DOMINATING AND LOCATING ...

RESOLVING, DOMINATING AND LOCATING DOMINATING SETS IN CO-NORMAL PRODUCT OF GRAPHS SHAHID UR REHMAN1 , IMRAN JAVAID1,∗ , MUHAMMAD IMRAN2,3

Abstract. A set W ⊆ V (G) is a resolving set for G if for any two distinct vertices of G not in W , there is a vertex in W which is at different distance from these vertices in G. A set D ⊆ V (G) is a dominating set for G, if every vertex of G not in D has at least one neighbor in D. A set S is a locating dominating set for G, if S is a dominating set and distinct vertices not belonging to S have distinct neighbors in S. The minimum cardinality of a resolving set, a dominating set and a locating dominating set of a graph G is called the metric dimension, domination number and location domination number of the graph G, respectively. Co-normal product of two graphs G and H is a graph having the vertex set V (G) × V (H) and the adjacency relation defined as (u, v) ∼ (x, w) if and only if u ∼ x in G or v ∼ w in H. In this article, we study resolving, dominating and locating dominating sets in co-normal product of two graphs. We provide bounds for the metric dimension, the domination number and the location domination number of co-normal product of graphs. We identify conditions on G and H for sharpness of these bounds. General formulae for the metric dimension of co-normal product of some standard families of graphs are also given.

1. Introduction Let G be a graph having the vertex set V (G) and the edge set E(G). Two vertices u, v in G are called adjacent if there is an edge between them, denoted by u ∼ v. The open neighborhood of a vertex v in G is denoted by NG (v) and NG (v) = {u ∈ V (G) : u ∼ v}. The closed neighborhood of a vertex u is denoted by NG [u] and NG [u] = NG (u) ∪ {u}. The cardinality of the open neighborhood of a vertex v ∈ V (G) is called the degree of v in G, denoted by degG (v). The diameter of a graph G, denoted by diam(G), is the maximum distance between any pair of vertices of G. If G is disconnected and two vertices u, v belongs to different components of G, then the distance between them is defined to be infinity and we write it as d(u, v) = ∞. A graph having no edge is called null graph. Two distinct vertices u, v are called true twins if N [v] = N [u] and f alse twins if N (v) = N (u). Two vertices are called twins if they are true or false twins. A subset Key words and phrases. Resolving set, adjacency resolving set, dominating set, locating dominating set, co-normal product. 2000 Mathematics Subject Classification. 05C12, 05C69 ∗ Corresponding author: [email protected]. 1

X of the vertices of a graph G is called independent set if no two vertices in X are adjacent. The cardinality of a maximum independent set in a graph G is called the independence number of G, denoted by α(G). Complement of an independent set in G is a vertex cover for G and the number β(G) = n − α(G) is called the vertex cover number of G. The clique number of a graph G is the cardinality of a maximum subset of vertices of G whose induced subgraph is a complete graph. A graph G is called nonempty graph if E(G) 6= φ. For concepts and terminology not defined in this paper, please see [5]. A set W ⊆ V (G) is a resolving(locating) set for G if for any two distinct vertices u, v ∈ V (G), there exist at least one vertex w ∈ W such that d(v, w) 6= d(u, w). The cardinality of a smallest resolving set of a graph G is called the metric dimension (location number) of G, denoted by dim(G)/loc(G). We will use the term metric dimension for making the symmetry with its variation named as the adjacency metric dimension defined below. This parameter has been studied in graph theory since 1975. Harary and Melter [22] and Slater [36] independently introduced the notion of the metric dimension of a graph. Applications of this parameter have appeared in different fields such as coin weighing problems [35], robot navigation [27], strategies for the Mastermind game [14], drug discovery [13], network discovery and verification [4]. Gary and Johnson [20] noted that the problem of finding the metric dimension of a graph is NP-hard, however, its explicit construction was given by Khuller et al. [27]. In graph theory, the concept of domination was initiated by Berge and Ore. This concept appears in our daily life problems such as: choosing a set of representatives, facility location problems, monitoring communication networks/electrical networks and land surveying. A set D ⊆ V (G) is a dominating set for 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). In 1958, Berge [5] introduced the concept of the coefficient of external stability, which Ore [30] re-introduced in 1962, with the name of domination number and gives its applications in real world. This parameter also plays an important role in on-line social networks [6] such as Facebook. It has been proved [20] that the problem of finding the domination number of an arbitrary graph is NP-complete. In [1], Bagheri et al. proved that dim(G) ≤ |G| − γ(G) and showed that equality holds if and only if G is a complete graph or a complete bipartite graph Kr,t , r, t ≥ 2. Both of these parameters have been widely investigated due to their applications in different fields. Motivated by a safeguard analysis problem, Slater [37, 38], introduced the concept of locating dominating set. A subset S is a locating dominating set for G if S is a dominating set and N (u) ∩ S 6= N (v) ∩ S for all u, v ∈ V (G)\S. The cardinality of a minimum locating dominating set is called the location domination number of G, denoted by λ(G). It was shown in [9] that λ(G) = 1 if and only if G = P2 and 2

|G| ≤ 5 if λ(G) = 2. Some simple relations such as γ(G) ≤ λ(G), dim(G) ≤ λ(G) and max{γ(G), dim(G)} ≤ λ(G) are also given in [9]. Caceres et al. [10] proved that λ(G) + d diam(G)+1 e ≤ |G|. For more results related to this parameter see 5 [29, 2, 18, 25, 12, 32]. P roduct graph of two graphs G and H is a new graph having the vertex set V (G)×V (H), the adjacency relation between the vertices can be defined in different ways using the adjacency relations of G and H so different product graphs can be obtained from G, H where G, H are called components of these graphs. It was given in [3] that there are 256 possible products between any two graphs G and H using the adjacency and the non adjacency relations of its components. In product graphs, there is a well known product graph introduced by Ore [30] in 1962, under the name Cartesian sum of graphs. It was named co−normal product graph by Hammack et al. [21]. Co−normal product (the terminology we have adopted) of graphs G and H with vertex sets V (G) = {v1 , v2 , . . . , vm } and V (H) = {u1 , u2 , . . . , un }, respectively, is the graph with the vertex set V (G) × V (H) = {vij = (vi , uj ) : vi ∈ V (G) and uj ∈ V (H)} and the adjacency relation defined as vij ∼ vrs if vi ∼ vr in G or uj ∼ us in H. Co-normal product graph obtained from graphs G and H is denoted by GH . For any two graphs G and H, the graph obtained from the vertex set V (G) × V (H), by defining the adjacency relation as (u, v) ∼ (x, y), if u ∼ x in G or u = x and v ∼ y in H, is called the lexicographic product graph, denoted as G[H]. G[H] ∼ = GH if and only if G is complete and H is arbitrary or G is arbitrary and H is null graph. Different properties and results regarding coloring of co-normal product graph were discussed in [19, 17]. Also, Yang [40] gave some results on the chromatic number of this product. For further work on this product, see [7, 15, 31]. Major task in product graphs is to discuss its structural properties in terms of structural properties of its components as in [17, 40]. Using the knowledge about the parameters of the components of product graphs, different parameters for product graphs have been discussed in [8, 15, 26, 33, 34]. In [34], Saputro et al. gave general bounds of the metric dimension of a lexicographic product of any connected graph G and an arbitrary graph H. Jannesari and Omoomi [26], use the adjacency metric dimension of factors of the lexicographic product graph to find the metric dimension of this product graph. All graphs considered in this paper are simple. In next section, we describe some structural properties of co-normal product of two graphs. Section 3 contains results on the metric dimension of co-normal product graph. We show that dim(GH ) = m·dim(H)+adim(G), if G = Pm , H = Sn where Pm is path graph having n vertices, Sn is star graph having n + 1 vertices. Also, we prove that dim(GH ) = |H|dim(G), if G = Km , H = Nn , where G = Km is complete graph of order m, H = Nn is null graph having order n. In section 4, we prove general results for the domination number of this product using the domination number of its components. In section 3

5, we prove results for the location domination number and describe bounds for the location domination number of co-normal product graph. 2. Co-normal Product of Graphs All results given in this paper for GH also hold for HG due to the commutativity of this product. Using the structures of the components of the co-normal product graph, we have some observations and known results related to structural properties of co-normal product graph. Observation 1. For any vertex vij ∈ V (GH ), deg(vij ) = |H|deg(vi ) + (|G| − deg(vi ))deg(uj ). Observation 2. For any vertex vij ∈ V (GH ), N (vij ) = N (vi ) × V (H) ∪ (N (vi ))c × N (uj ). Observation 3. For any two integers m, n ≥ 2, if G = Km and H = Nn , then GH = Kn,n,...,n with m partite. Observation 4. For any two connected graphs G and H, the product graph GH has girth 3. Theorem 2.1. [19] GH is connected if and only if one of the following holds: (1) H = Kn for some n ≥ 2 and G is connected. (2) G = Km for some m ≥ 2 and H is connected. (3) G and H are nonempty and at least one of G or H is without isolated vertices. Theorem 2.2. [28] 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(GNn ) = max{2, diam(G)}. (2) G and H have isolated vertices, then diam(GH ) = ∞. (3) If neither G nor H has isolated vertices, then diam(GH ) = 2. (4) If diam(H) ≤ 2, then diam(GH ) = 2. (5) If diam(H) > 2, H has no isolated vertices and G is a non-empty graph having at least one isolated vertex, then diam(GH ) = 3. Lemma 2.3. For any two distinct vertices vij and vrs in GH , N (vij ) = N (vrs ) if and only if 1) vi = vr in G and N (uj ) = N (us ) in H, or 2) uj = us in H and N (vi ) = N (vr ) in G or 3)N (vi ) = N (vr ) in G and N (uj ) = N (us ). Proof. By observation 2, for any vertex vij ∈ V (GH ), N (vij ) = N (vi ) × V (H) ∪ (N (vi ))c ×N (uj ). If N (vij ) = N (vrs ) in GH we have N (vi )×V (H)∪(N (vi )c ×N (uj ) = N (vr ) × V (H) ∪ (N (vr )c × N (us ), then we have following possibilities: 1) vi = vr in G and N (uj ) = N (us ) in H, or 2) uj = us in H and N (vi ) = N (vr ) in G or 3)N (vi ) = N (vr ) in G and N (uj ) = N (us ). ¤ A subset R of the vertices of a graph G is called a packing set for G if N [x] ∩ N [y] = φ for all x, y ∈ R. For every vi ∈ V (G) and uj ∈ V (H), the classes 4

C(vi ) = {vk ∈ V (G)|N (vi ) = N (vk )} and C(uj ) = {ul ∈ V (H)|N (uj ) = N (ul )} are equivalence classes of false twins in G and H, respectively. We define vertex sets, H(vi ) = {vij : uj ∈ V (H)} ⊆ V (GH ) for vi ∈ V (G) and G(uj ) = {vij : vi ∈ V (G)} ⊆ V (GH ) for uj ∈ V (H). For any vertex vij ∈ V (GH ), the class C(vij ) = {vkl ∈ V (GH )|N (vkl ) = N (vij )}, is an equivalence class of vij in GH . Observation 5. For any vertex vij ∈ V (GH ), we have C(vij ) = C(vi ) × C(uj ), where C(vi ), C(uj ) are equivalence classes in G and H, respectively. Observation 6. For any two non trivial connected graphs G and H, the product graph GH has no packing subset. Proposition 2.4. If W1 is a maximum independent set for G and W2 is a maximum independent set for H, then W1 × W2 is a maximum independent set for GH . Proof. Since for any two vertices vi , vk ∈ W1 , vi  vk in G and for any two vertices uj , ul ∈ W2 , we have uj  ul in H. So, for vij , vkl ∈ W1 × W2 , vij  vkl in GH ´ is a maximum which gives that W1 × W2 is an independent set for GH . Suppose W ´ | > |W |, then there must exists at least one independent set of GH such that |W ´ . Also, the induced vr ∈ V (G) such that |W (vr )| > |W2 |, where W (vr ) = H(vi ) ∩ W subgraph of H(vr ) is isomorphic to H. Hence, W (vr ) is not an independent set in GH , a contradiction. ¤ Proposition 2.5. For any two distinct vertices vij , vkl in co-normal product of two nontrivial connected graphs G and H, we have N (vij ) ∩ N (vkl ) 6= φ. Proof. Let vi = vk so there exists vr ∈ V (G) such that vi ∼ vr because G is connected. This gives that H(vr ) ⊂ N (vij ) ∩ N (vkl ). Suppose that, uj = ul , then G(us ) ⊂ N (vij ) ∩ N (vkl ) for some us ∈ V (H) because H is connected. If vi 6= vk and uj 6= ul , then there exists us ∈ V (H) and vr ∈ V (G) such that vr ∼ vk and us ∼ uj which gives that vrs ∈ N (vij ) ∩ N (vkl ). ¤ 3. Metric Dimension of Co-normal Product of Graphs For any vertex vi ∈ V (G) and W = {w1 , w2 , . . . , wk } ⊂ V (G), the k-vector (d(vi , w1 ), d(vi , w2 ), . . . , d(vi , wk )) is called the metric representation of vi with respect to ordered set W , denoted by γ(vi |W ). W is called a resolving set for G if the metric representations of any two distinct vertices with respect to W are different. The cardinality of a minimum resolving set for G is called the metric dimension of G, denoted by dim(G). For any two vertices u, v ∈ V (G), the adjacency relation between them is defined as:   0 if u = v, a(u, v) = 1 if u ∼ v,  2 if u  v. The adjacency representation of a vertex vi ∈ V (G) with respect to an ordered set W = {w1 , w2 , . . . , wk } is defined as γ2 (vi |W ) = (a(vi , w1 ), a(vi , w2 ), . . . , a(vi , wk )). A 5

subset W ⊆ V (G) is an adjacency resolving set [26] for G, if γ2 (vi |W ) 6= γ2 (vj |W ) for any two distinct vertices vi , vj in G. The cardinality of a minimum adjacency resolving set for G, is called the adjacency metric dimension of G, denoted by adim(G). Jannesari and Omoomi introduced this notion and use it to find the bounds for the metric dimension of lexicographic product graph. In this section, we describe results for resolving sets and adjacency resolving sets of co-normal product graph. We also find some bounds for the metric dimension of co-normal product graph using the order and adjacency metric dimension of this product. It is clear that dim(G) = adim(G), if G is a connected graph having diameter 2. Note that in case of co-normal product of any two nontrivial connected graphs, the adjacency metric dimension is same as the metric dimension. The set symmetric dif f erence of two sets A and B is defined as (A ∪ B) \ (A ∩ B), denoted by A M B. Using the definition of symmetric difference of two sets we have following proposition: Proposition 3.1. If G is a connected graph having diameter 2 and W is a minimum resolving set for G, then |W ∩ S{u, v}| ≥ 1 for all u, v ∈ V (G), where S{u, v} = {u, v} ∪ (N (u) M N (v)). Lemma 3.2. For any ordered subset W (vi ) of H(vi ) if vij ∈ / W (vi ), then for any vk , vl  vi , γ(vij |W (vi )) = γ(vkj |W (vi )) = γ(vlj |W (vi )). Proof. Let vim ∈ H(vi ), if vij ∼ vim in GH , then uj ∼ um in H so vkj ∼ vim in GH and if vij  vim in GH , then uj  um in H so vkj  vim in GH . Hence, for any ordered subset W (vi ) of H(vi ) we have γ(vkj |W (vi )) = γ(vij |W (vi )). Also, for any two vertices vk , vl  vi if vij ∈ / W (vi ), then γ(vkj |W (vi )) = γ(vij |W (vi )) and γ(vlj |W (vi )) = γ(vij |W (vi )) so γ(vkj |W (vi )) = γ(vlj |W (vi )). ¤ Lemma 3.3. A subset W of vertices of GH is a resolving set for GH if and only if for any two distinct vertices vij , vrs , there exists at least one vertex vl in G, such that N (vij ) ∩ W (vl ) 6= N (vrs ) ∩ W (vl ), where W (vl ) = W ∩ H(vl ). Proof. Let W be a resolving set for GH . Assume contrary that there exist two vertices vij , vrs in GH such that for every vl in G, we have N (vij ) ∩ W (vl ) = N (vrs ) ∩ W (vl ), then for every vl in G, γ(vij |W (vl )) = γ(vrs |W (vl )). Because the code is the adjacency representation of vij , vrs with respect to the vertex set W (vl ). Since, W = ∪vl ∈V (G) W (vl ), hence γ(vij |W ) = γ(vrs |W ), a contradiction. Conversely, let W be a subset of vertices of GH such that for any two distinct vertices vij , vrs of GH , there exists at least one vertex vl in G such that N (vij ) ∩ W (vl ) 6= N (vrs ) ∩ W (vl ), then γ(vij |W (vl )) 6= γ(vrs |W (vl )) and γ(vij |W ) 6= γ(vrs |W ). ¤ Next two theorems give conditions under which there exists resolving set W for GH such that W ∩ H(vi ) = φ for any vi ∈ V (G) and W ∩ G(uj ) = φ for any uj ∈ V (H). 6

Theorem 3.4. There exist a resolving set W for GH such that W ∩ H(vi ) = ∅ if and only if for any two distinct vertices uj , us in H, we have N (uj ) 6= N (us ). Proof. Suppose there exists a resolving set W for GH such that W ∩ H(vi ) = φ, we are to show that there does not exist any two distinct vertices uj , us in H for which N (uj ) = N (us ). Assume contrary that there are two distinct vertices in H such that N (uj ) = N (us ), then by using Lemma 2.3, N (vij ) = N (vis ) in GH . So, for every vl in G, N (vij ) ∩ W (vl ) = N (vrs ) ∩ W (vl ). Hence, W is not a resolving set for GH . Conversely, suppose that vl  vi in G and the set W (vl ) ⊆ H(vl ) such that N (vij ) ∩ W (vl ) 6= N (vis ) ∩ W (vl ) so γ(vij |W (vl )) 6= γ(vis |W (vl )). Hence, we can choose W such that W (vl ) ⊆ W and W ∩ H(vi ) = φ ¤ Lemma 3.5. If W2 is an adjacency resolving set for H, then for any vi ∈ V (G), the vertices of H(vi ) are resolved by its subset W (vi ) = {vi } × W2 . Proof. We prove that W (vi ) resolves the vertices of H(vi ). Let vij , vik ∈ H(vi )\W (vi ), then there exists vit ∈ W (vi ) such that |N (vit ) ∩ {vij , vik }| = 1 because the vertices of H(vi ) preserves the adjacency relation of H in GH and W2 is adjacency resolving set for H. ¤ Theorem 3.6. Let G be a twin free graph having order m and C(u1 ), C(u2 ), . . . , C(uk ) be the distinct equivalence classes in H with the property that | C(ui ) |6= 1, for each 1 ≤ i ≤ k, then dim(GH ) = m · dim(H). Proof. Since G have no twins so N (vi ) 6= N (vk ), for vi 6= vk ∈ V (G). This implies that G have m distinct equivalence classes. Observation 5 gives that the co-normal product GH has mk equivalence classes such that no class has cardinality one, so i=m P j=k P |C(vij )|− mk also |C(vij )| = |C(uj )| for each vi ∈ V (G) and dim(GH ) = i=1 j=1

uj ∈ {u1 , u2 , . . . , uk } which gives that dim(GH ) =

i=m P j=k P

|C(uj )|− mk.

¤

i=1 j=1

Corollary 3.7. If G = Pm ; m ≥ 4 and H = Kn1 ,n2 ,...,nl , then dim(GH ) = m

j=l Q

(nj −

j=1

1). In the following theorem, we give answer to the question that if W2 is an adjacency resolving set for H and W (vi ) = {vi } × W2 , then under what conditions W = ∪vi ∈V (G) W (vi ) is a resolving set for GH . Theorem 3.8. Let G be a graph of order m having no false twins and H be any graph. If there exists an adjacency basis W2 of H such that, γ2 (uj |W2 ) 6= (1, 1, . . . , 1) for all uj ∈ V (H), then dim(GH ) ≤ m · adim(H). Proof. Let W (vi ) = {vi } × W2 and W = ∪vi ∈V (G) W (vi ) or W = V (G) × W2 . By Lemma 3.5, W (vi ) resolves all the vertices of H(vi ). To show that W is an adjacency 7

resolving set for GH , consider two distinct vertices vij , vkl ∈ V (GH )\W . We discuss following cases: Case 1: Let vi = vk and uj , ul ∈ V (H)\W2 . Since, W2 is an adjacency resolving set for H, so N (uj ) ∩ W2 6= N (ul ) ∩ W2 . Hence, N (vij ) ∩ W 6= N (vkl ) ∩ W . Case 2: Let uj = ul also N (vi ) ∩ V (G) 6= N (vk ) ∩ V (G) for all vi , vk ∈ V (G). Hence, N (vij ) ∩ W 6= N (vkl ) ∩ W . Case 3: If vi 6= vk and uj 6= ul , then N (vi )∩V (G) 6= N (vk )∩V (G) and N (uj )∩W2 6= N (ul ) ∩ W2 . Hence, N (vij ) ∩ W 6= N (vkl ) ∩ W . ¤ Corollary 3.9. Let G be a complete graph and H 6= Kn be an arbitrary graph. If H has an adjacency basis W2 such that, γ(uj |W2 ) 6= (1, 1, . . . , 1) for all uj ∈ V (H), then dim(GH ) = m · adim(H). Proof. Since, G is complete so G have no false twins. Also, W2 satisfies the condition of Theorem 3.8, so dim(GH ) ≤ m · adim(H). Now for some vi ∈ V (G), uj ∈ W2 , consider W = V (G) × (W2 \{uj }) and note that W (vi ) = {vi } × (W2 \{uj }) will not resolves the vertices of H(vi ), because W2 is an adjacency basis of H, so there exists ul ∈ V (H)\W2 such that γ(ul |W2 \{uj }) = γ(uj |W2 \{uj }) which gives that γ(vil |W (vk )) = γ(vij |W (vk )) for all vk 6= vi because G is complete. Hence, dim(GH ) ≥ m · adim(H). ¤ Theorem 3.10. Let G be a complete graph and H 6= Kn be an arbitrary graph. If for each adjacency basis W2 of H, there exist a vertex uj ∈ V (H)\W2 such that γ2 (uj |W2 ) = (1, 1, . . . , 1), then dim(GH ) = m(adim(H) + 1) − 1. Proof. By using Lemma 3.5, W (vi ) = {vi } × W2 will resolve the vertices of H(vi ). Since G is complete, γ(vij |W (vk )) = (1, 1, . . . , 1) for all vk 6= vi . Also γ(vij |W (vi )) = (1, 1, . . . , 1) for each vi ∈ V (G). Hence, W = ∪vi ∈V (G) W (vi ) is not a resolving set for GH . Also the induced subgraph of the vertex set G(uj ) = {vij |vi ∈ V (G)} is isomorphic to G and G is complete. Hence, dim(GH ) = madim(H) + m − 1. ¤ Since, GH is complete if and only if G, H are complete and GH is null graph if and only if G, H are null graphs. So, we have following straight forward result: Theorem 3.11. For any two graphs G and H of order m, n ≥ 2 respectively, adim(GH ) = mn − 1 if and only if both G and H are complete or both are null graphs. Theorem 3.12. Let G be a nontrivial connected graph and H 6= Kn be an arbitrary graph, then adim(H) · adim(G) ≤ dim(GH ) ≤ m · adim(H) + n · adim(G). Proof. Let W = W1 × V (H) ∪ V (G) × W2 , where W1 , W2 are adjacency basis of G and H respectively. Let Wvi = W ∩ H(vi ) for vi ∈ V (G), Wuj = W ∩ 8

G(uj ) for uj ∈ V (H). For any vertex vij ∈ V (GH ), the metric representation is of the form γ(vij |W ) = (γ2 (vij |Wv1 ), γ2 (vij |Wv2 ), . . . , γ2 (vij |Wvm )) or γ(vij |W ) = (γ2 (vij |Wu1 ), γ2 (vij |Wu2 ), . . . , γ2 (vij |Wun )). For any two distinct vertices vij , vkl ∈ V (GH )\W , we have vi , vk ∈ / W1 and uj , ul ∈ / W2 . To prove that W is a resolving set for GH , we discuss following cases: Case 1: Suppose vi = vk and note that the set W (vi ) = {vi }×W2 ⊆ Wvi . By Lemma 3.4, W (vi ) resolves the vertices of H(vi ) which gives that γ(vij |Wvi ) 6= γ(vkl |Wvi ). Hence, γ(vij |W ) 6= γ(vkl |W ). Case 2: Suppose uj = ul and note that the set W (uj ) = W1 ×{uj } ⊆ Wuj and W (uj ) resolves the vertices of G(uj ) which gives that γ(vij |Wuj ) 6= γ(vkl |Wuj ). Hence, γ(vij |W ) 6= γ(vkl |W ). Case 3: Let vi 6= vk and uj 6= ul . As, γ(vij |Wvi ) 6= γ(vil |Wvi ), γ(vkj |Wvk ) 6= γ(vkl |Wvk ) and γ(vij |Wuj ) 6= γ(vkj |Wuj ). Hence, γ(vij |W ) 6= γ(vkl |W ). Now for any two adjacency basis W1 and W2 of G and H, respectively. Let W = W1 × W2 where W2 is an adjacency basis of H so for any uj ∈ W2 , W2 \{uj } is not an adjacency resolving set for H. Thus, there exist ul ∈ V (H)\W2 such that, N (uj ) ∩ W2 \{uj } = N (ul ) ∩ W2 \{uj }. Since, vij , vil ∈ V (GH )\W for some vi ∈ V (G)\W1 and H(vi ) ∩ W = φ, so for every vk 6= vi in G, γ(vij |W ) = γ(vil |W ). Hence, W is not a resolving set for GH . ¤ Theorem 3.13. If G is a complete graph and H is a null graph having order m, n ≥ 2 respectively, then. dim(GH ) = m(n − 1). Proof. Let V (G) = {v1 , v2 , . . . , vm } and V (H) = {u1 , u2 , . . . , un }. It is clear from the definition of co-normal product that for each vi , N (vij ) = N (vik ) for all 1 ≤ j, k ≤ n. So any resolving set must contain at least n − 1 vertices from each H(vi ) which gives that dim(GH ) ≥ m(n−1). Since, H is null graph so γ(vij |H(vi )\{vij }) = (2, 2, . . . , 2) for each i and γ(vij |H(vk )) = (1, 1, . . . , 1) for each k 6= i which gives that any subset of V (GH ) containing n − 1 vertices from each H(vi ) will be a resolving set for GH . Hence, dim(GH ) = m(n − 1). ¤ Theorem 3.14. 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(GH ) = m·dim(H)+adim(G). Proof. Let V (G) = {v1 , v2 , . . . , vm } and V (H) = {u0 , u1 , u2 , . . . , un }, where deg(u0 ) = n in H. Also, N (uk ) = N (ul ) for all 1 ≤ k, l ≤ n, by using Lemma 2.3, we have N (vik ) = N (vil ) for each i. So, any resolving set W for G must contain at least n − 1 vertices from each H(vi ). Since, deg(u0 ) = n so by the definition of co-normal product d(vi0 , vij ) = 1 for all 1 ≤ i ≤ m and 1 ≤ j ≤ n, which means that the vertices of G(u0 ) are not resolved by any of vij , 1 ≤ i ≤ m, 1 ≤ j ≤ n. Also, d(vi0 , vj0 ) ≤ 2 in GH and induced subgraph of G(u0 ) is isomorphic to G so we must choose adim(G) vertices from G(u0 ), which gives that dim(GH ) = m · dim(H) + adim(G). ¤ 9

4. Domination in Co-normal Product A subset W of the vertices of G is a dominating set for G, if ∪u∈W N [u] = V (G). The cardinality of a minimum dominating set for G is called the domination number of G, denoted by γ(G). A vertex v is called a dominating vertex in G if N [v] = V (G). Lemma 4.1. A vertex vij is a dominating vertex in GH if and only if vi and uj are dominating vertices in G and H respectively. Proof. Let vij be a dominating vertex in GH . To show that vi , uj are dominating in G and H respectively, assume contrary that vi is not dominating in G so there exists vk ∈ V (G) such that vk ∈ / N (vi ), then vkj ∈ / N (vij ) a contradiction. Now suppose that vi and uj are dominating vertices in G and H respectively then by the observation 1, we have deg(vij ) = mn − 1. ¤ Lemma 4.2. If G has a dominating vertex and H has no dominating vertex, then γ(GH ) = 2. Proof. Suppose vi is a dominating vertex of G, so using the definition of co-normal product, vij ∼ vkl for all vkl ∈ V (GH ) with vi 6= vk . Also, H has no dominating vertex so there must be a vertex ur ∈ V (H) such that ur ∈ / N (uj ) which gives that vir ∈ / N (vij ). Now for any vertex vk ∼ vi , the set {vij , vkl }, is a dominating set for GH for any chosen vertex vkl ∈ H(vk ). Hence, γ(GH ) = 2. ¤ Lemma 4.3. If W1 and W2 are dominating sets for G and H respectively, then W1 × W2 is a dominating set for GH . Proof. Let W1 = {´ v1 , v´2 , . . . , v´n1 }, W2 = {´ u1 , u´2 , . . . , u´n2 } be dominating sets for G, H, respectively and W = W1 × W2 . To show that W is a dominating set for GH , consider a vertex vij ∈ V (GH ), we have following cases: Case 1: If vi ∈ W1 and uj ∈ W2 , then vij ∈ ∪vij ∈W N [vij ]. Case 2: If vi ∈ W1 and uj ∈ / W2 , then there exists uk ∈ W2 such that uj ∈ N (uk ) also vik ∈ W so vij ∈ N (vik ). Case 3: If vi ∈ / W1 and uj ∈ W2 , then there exists vk ∈ W1 such that vi ∈ N (vk ) also vkj ∈ W so vij ∈ N (vkj ). Case 4: Let vi ∈ / W1 and uj ∈ / W2 , then there exists vk ∈ W1 and ul ∈ W2 such that vi ∈ N (vk ) and uj ∈ N (ul ) so vij ∈ N (vkl ) for vkl ∈ W . Hence, W is a dominating set for GH . ¤ A set D ⊆ V (G) is called a total dominating set for G [16], if D is a dominating set and for each v ∈ D, there exist u ∈ v(G) such that v ∼ u in G. The total domination number of G, denoted by γt (G), is the cardinality of a minimum total dominating set for G. 10

Theorem 4.4. For any two graphs G and H with γ(G), γ(H) ≥ 2 and γ(G) ≤ γ(H), if γ(G) = γt (G), then γ(GH ) = γ(G). Proof. Let W1 = {´ v1 , v´2 , . . . , v´n1 } be a minimum dominating set for G such that for each v´i ∈ W1 , there exists v´j ∈ W1 such that N (´ vi ) ∩ W1 6= φ in G. Consider W = {´ v11 , v´22 , . . . , v´n1 n1 } ⊆ V (GH ) where v´ii = (´ vi , u´i ), v´i ∈ W1 and u´i ∈ V (H). ´ = ∪v´ ∈W N [´ First we show that W is a dominating for GH . Let W vii ] ⊆ V (GH ) and ii let vij ∈ V (GH ) so we have following cases: ´. Case 1: If vij ∈ W , then vij ∈ W Case 2: If vij ∈ / W with vi ∈ W1 , then there exists v´k ∈ W1 such that vi ∼ v´k so ´ ´ . Hence, W vij ∈ W . For vi ∈ / W1 we have v´k ∈ W1 such that vi ∈ N (´ vk ) so vij ∈ W is dominating set for GH . Now to prove that W is a minimum dominating set, assume contrary that W is not a minimum dominating set for GH and let D = {´ vij |´ vij ∈ V (GH )} be a dominating set for GH such that |D| < γ(G), then the sets W1 = {´ vi ∈ V (G) | v´ij ∈ D f or some u´j ∈ V (H)}, W2 = {´ uj ∈ V (H) | v´ij ∈ D f or some v´i ∈ V (G)} are not the dominating sets for G and H respectively, so there exists vk ∈ V (G)\W1 and ul ∈ V (H)\W2 such that N (vi ) ∩ W1 = φ, N (ul ) ∩ W2 = φ so N (vkl ) ∩ D = φ, a contradiction. ¤ Theorem 4.5. For arbitrary graphs G and H, min{γ(G), γ(H)} ≤ γ(GH ) ≤ γ(G)γ(H). Proof. The upper bound directly follows from Lemma 4.3. For lower bound, consider γ(G) ≥ 1 and γ(H) ≥ 1, first suppose that γ(G) = 1 and γ(H) = 1, then by Lemma 4.1, γ(GH ) = 1. If γ(G) = 1 and γ(H) ≥ 2, then by Lemma 4.2, we have γ(GH ) = 2. Now, if γ(G) ≥ 2, γ(H) ≥ 2 and γ(G) ≤ γ(H) such that G has a minimum dominating set which satisfies the conditions of Theorem 4.4, then γ(GH ) = γ(G) and if G has no minimum dominating set satisfying the conditions of Theorem 4.4, then for any minimum dominating set W1 = {´ v1 , v´2 , . . . , v´m1 } of G we have at least one vertex v´i ∈ W1 such that v´i  v´k for all v´k ∈ W1 . Now for any subset W2 of V (H) such that | W2 |≤ γ(H). Consider the subset W = {´ v11 , v´22 , . . . , v´m1 m1 } ⊆ V (GH ) where u´i ∈ W2 , we discuss following cases: Case 1: If | W2 |= γ(H) and W2 is a dominating set for H, then γ(GH ) = γ(H) = min{γ(G), γ(H)}. Case 2: If W2 is not a dominating set for H, then there exists u´j ∈ V (H) such that N (´ uj ) ∩ W2 = φ, N (´ vij ) ∩ W = φ. Hence, W is not a dominating set for GH and γ(GH ) ≥ min{γ(G), γ(H)}. ¤ 5. Location Domination in Co-normal Product A locating dominating set D of a graph G, is a dominating set of G having the property that NG (x) ∩ D 6= NG (y) ∩ D, for any two distinct vertices x, y ∈ V (G)\D. 11

The location domination number of G, denoted by λ(G), is the minimum cardinality of a locating dominating set in G. Following proposition directly follows from the definition of set symmetric difference and location domination number of a connected graph G. Proposition 5.1. If D is locating dominating set for G, then |D ∩ S{u, v}| ≥ 1 for all u, v ∈ V (G) \ D such that N (u) ∩ N (v) 6= ∅ and |D ∩ S{u, v}| ≥ 2 for all u, v ∈ V (G) \ D such that N (u) ∩ N (v) = ∅, where S{u, v} = N (u) M N (v). In next theorem, we give conditions for any connected graph under which the three parameters metric dimension, adjacency metric dimension and location domination number are equal for the graph. Theorem 5.2. If G is a connected graph having k distinct equivalence classes C(v1 ), C(v2 ),..., C(vk ) with |C(vi )| = mi , 1 ≤ i ≤ k and mi ≥ 2, then dim(G) = adim(G) = λ(G). Proof. Let W1 = {v1 , v2 , . . . , vk } where v1 , v2 , . . . , vk belongs to different equivalence classes of G. For any vertex v ∈ V (G) the induced subgraph < C(v) > is a null graph, also N (vi ) ∩ W1 6= φ because for N (vi ) ∩ W1 = φ, G will be disconnected. Consider W = V (G)\W1 , we show that W is locating dominating set for G. By using the fact, N (vi ) ∩ W1 6= φ for each 1 ≤ i ≤ k and d(vi , vj ) = d(vi , C(vj )), we have N (vi ) ∩ W 6= φ. Now, we only need to prove that N (vi ) ∩ W 6= N (vj ) ∩ W for any two distinct vertices vi , vj ∈ W1 . If vi ∼ vj , then N (vi ) ∩ W 6= N (vj ) ∩ W because C(vi )\{vi } ⊆ W so C(vi )\{vi } ⊆ N (vj ) ∩ W but C(vi )\{vi } * N (vi ). Now suppose that, vi  vj and N (vi ) ∩ W = N (vj ) ∩ W for any two distinct vertices vi , vj ∈ W1 . For each vi ∈ W1 , let W (vi ) = {vk ∈ W1 | vk ∼ vi } ⊂ W1 , then N (vi ) = ∪vk ∈W (vi ) C(vk ) and N (vi ) ∩ W = ∪vk ∈W (vi ) C(vk )\W1 , so N (vi ) = N (vj ), a i=k i=k P P mi − k. Also, dim(G) ≥ mi − k because G have contradiction. Thus, λ(G) ≤ i=1

i=1

k distinct equivalence classes. Hence, dim(G) = adim(G) = λ(G).

¤

Corollary 5.3. If G = Pm ; m ≥ 4 and H = Kn1 ,n2 ,...,nl where ni ≥ 2 for each i, then λ(GH ) = adim(GH ) = dim(GH ). Theorem 5.4. [11] Let G be a connected graph such that adim(G) < λ(G). Then, 1 + adim(G) = λ(G). By Theorem 5.4 and using the definition of locating dominating set, we have following bounds: adim(GH ) ≤ λ(GH ) ≤ 1 + adim(GH ) Theorem 5.5. [39] If G is a regular graph of degree r, then λ(G) ≥

2|G| . r+3

GH is regular if and only if both G, H are regular and GH is complete if and only if G, H are complete. Since, for any r − regular graph G and s − regular graph H, 12

GH is (ms + nr − rs) − regular graph so by Theorem 5.5, we have following bounds: 2mn ≤ λ(GH ) ≤ mn − 1 ms + nr − rs + 3 An upper bound for the location domination number of the co-normal product of two graphs G and H can be seen in next theorem. Theorem 5.6. Let D1 , D2 be locating dominating sets for G and H respectively, then the set D = V (G) × D2 ∪ D1 × V (H) is a locating dominating set for GH . Proof. For any two distinct vertices vij , vkl ∈ V (GH )\D, we have vi , vk ∈ / D1 and uj , u l ∈ / D2 . To show that D is locating dominating set for GH we discuss following cases: Case 1: If vi = vk and uj 6= ul , then there exists ut ∈ D2 such that ut ∈ N (uj ) and ut ∈ / N (ul ). Also, vit ∈ D, vit ∈ N (vij ) and vit ∈ / N (vkl ). Case 2: If vi 6= vk and uj = ul , then there exists vr ∈ D1 such that vr ∈ N (vi ) and vr ∈ / N (vk ). Also, vrj ∈ D, vrj ∈ N (vij ) and vrj ∈ / N (vkl ). Case 3: If vi 6= vk and uj 6= ul , then there exists vr ∈ D1 such that vr ∈ N (vi ), vr ∈ / N (vk ) and ut ∈ D2 such that ut ∈ N (ui ) and ut ∈ / N (ul ). Also, vrj ∈ D, vrj ∈ N (vij ) but vrj ∈ / N (vkl ) and vit ∈ D, vit ∈ N (vij ) but vit ∈ / N (vkl ). From the definition of D and using the structure of GH , for any vertex vij ∈ V (GH )\D we have N (Vij ) ∩ D 6= φ. Hence, D is a locating dominating set for GH . ¤ From Theorem 5.6, we have λ(GH ) < nλ(H)+mλ(G). Also, for any two minimum locating dominating sets D1 , D2 of G and H respectively. Let D = D1 × D2 , vi ∈ V (G)\D1 and vk ∈ D1 such that N (vi ) ∩ D1 \{vk } = N (vk ) ∩ D1 \{vk }, then N (vij ) ∩ D = N (vij ) ∩ D. Hence, we have λ(G)λ(H) < λ(GH ) < nλ(H) + mλ(H). Theorem 5.7. Let G be a twin free graph. If H has minimum locating dominating set D2 having the property that D2 * N (uj ) for all uj ∈ V (H)\D2 , then D = V (G) × D2 is a locating dominating set for GH . Proof. For any two distinct vertices vij , vkl ∈ V (GH )\D, we have uj , ul ∈ / D2 . To show that D is a locating dominating set for GH we discuss following cases: Case 1: If vi = vk and uj 6= ul , then there exists ut ∈ D2 such that ut ∈ N (uj ) and ut ∈ / N (ul ). Also, vit ∈ D, vit ∈ N (vij ) and vit ∈ / N (vkl ). Case 2: If vi 6= vk and uj = ul , then there exists vr ∈ V (G) such that vr ∈ N (vi ) and vr ∈ / N (vk ) because G is twin free. Since D2 * N (uj ) for all uj ∈ / V (H)\D2 so there exists at least one ut ∈ D2 such that ut ∈ / N (uj ). Also, vrt ∈ D, vrt ∈ N (vij ) and vrt ∈ / N (vkl ). Case 3: If vi 6= vk and uj 6= ul , then there exists vr ∈ V (G) such that vr ∈ N (vi ), vr ∈ / N (vk ) and ut ∈ D2 such that ut ∈ N (uj ) and ut ∈ / N (ul ). Also, vrt ∈ D, vrt ∈ N (vij ) but vrt ∈ / N (vkl ) and vit ∈ D, vit ∈ N (vij ) but vit ∈ / N (vkl ). From the definition of D and using the structure of GH it can be verified that for any vertex 13

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Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan. 2 Department of Mathematical Sciences, United Arab Emirates University, P. O. Box 15551, Al Ain, United Arab Emirates. 3 Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, Islamabad, Pakistan. Email: [email protected], [email protected], [email protected]

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