Roman domination in direct product graphs and rooted product graphs

Let $G$ be a graph with vertex set $V(G)$. A function $f:V(G)\rightarrow \{0,1,2\}$ is a Roman dominating function on $G$ if every vertex $v\in V(G)$ for which $f(v)=0$ is adjacent to at least one vertex $u\in V(G)$ such that $f(u)=2$. The Roman domination number of $G$ is the minimum weight $\omega(f)=\sum_{x\in V(G)}f(x)$ among all Roman dominating functions $f$ on $G$. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for the Roman domination number of direct product graphs involving some parameters of the factors, which include the domination, (total) Roman domination, and packing numbers among others. On the other hand, we prove that the Roman domination number of rooted product graphs can attain only three possible values, which depend on the order, the domination, and the Roman domination numbers of the factors in the product. In addition, theoretical characterizations of the classes of rooted product graphs achieving each of these three possible values are given.


Introduction
. The domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set of G. A dominating set of G with minimum cardinality is called a γ(G)-set. More information on domination in graphs can be found in the books [13,14]. In the last two decades, dominating functions have been extensively studied. One of the reasons may be due to the fact that dominating functions generalize the concept of dominating sets. A function f : V (G) → {0, 1, . . .} on G is said to be a dominating function if for every vertex v such that f (v) = 0, there exists a vertex u ∈ N(v), such that f (u) ≥ 1. If we restrict ourselves to the case of functions f : V (G) → {0, 1, 2}, then we observe that f generates three sets V 0 , V 1 and V 2 ; where V i = {v ∈ V (G) : f (v) = i} for i ∈ {0, 1, 2}. In such a sense, we will write f (V 0 , V 1 , V 2 ) to refer to the function f . Given a set S ⊆ V (G), f (S) = v∈S f (v). The weight of f is ω(f ) = f (V (G)) = |V 1 | + 2|V 2 |.
The theory of Roman dominating functions is one of the most studied topics within the theory of dominating functions in graphs. Roman dominating functions were formally defined by Cockayne et al. [7] motivated, in part, by a paper of Ian Stewart entitled "Defend the Roman Empire" [22]. A Roman dominating function (RDF) on a graph G is a function f (V 0 , V 1 , V 2 ) such that for every vertex v ∈ V 0 , there exists a vertex u ∈ N(v) ∩ V 2 . The Roman domination number of G, denoted by γ R (G), is the minimum weight ω(f ) = v∈V (G) f (v) among all RDFs f on G. Some results on Roman domination in graphs can be found for example, in [5,7,8,20,27].
Moreover, one of the most attractive research approaches within the domination theory in graphs is the study of domination-related parameters in product graphs. As expected, some of the researches concerning Roman dominating functions are related to product graphs. In particular, we cite the following works: lexicographic product graphs [3,18]; Cartesian and strong product graphs [24,26], direct product of paths and cycles [16,17], rooted product graphs [19], and corona product graphs [25].
It is now our goal to continue with the study of this parameter in two of the product graphs mentioned above. In Section 2 we obtain tight bounds for the Roman domination number of direct product graphs. In Section 3 we provide closed formulas for the Roman domination number of rooted product graphs, and characterize the graphs reaching these expressions. We end this present section with some terminology on invariants related to domination which will be further used.
A natural lower bound for γ(G) is the packing number of G. A set A ⊆ V (G) is called a packing set (or simply a packing) if any two distinct vertices x, y ∈ A satisfy that N[x] ∩ N[y] = ∅, that is, the closed neighborhoods of vertices in A have pairwise empty intersections. The cardinality of a largest packing in G is called the packing number and is denoted by ρ(G). Any packing of cardinality ρ(G) is called a ρ(G)-set. The open packing number is a natural lower bound for the total domination number γ t (G). This is the minimum cardinality of a set D ⊆ V (G), called a total dominating set, where every vertex of G has a neighbor in D. A total dominating set of cardinality γ t (G) is called a γ t (G)-set as usual.
The last invariant we mention here is the total Roman domination number γ tR (G). An RDF f (V 0 , V 1 , V 2 ) is called a total Roman dominating function (TRDF) if G[V 1 ∪ V 2 ] has no isolated vertices. The minimum weight ω(f ) = |V 1 | + 2|V 2 | among all TRDFs f (V 0 , V 1 , V 2 ) is called the total Roman domination number γ tR (G) on G. Again, we simply call a TRDF f of weight γ tR (G) as a γ tR (G)-function.
For some extra information (main results, open problems, etc.) on several domination related invariants, including the above mentioned ones, we suggest the recent books [11,12,15].

Direct product graphs
Let G and H be two graphs. Their direct product G × H is a graph with vertex set V (G) × V (H) and two vertices (u, v) and (u ′ , v ′ ) are adjacent in G × H if uu ′ ∈ E(G) and vv ′ ∈ E(H). The direct product belongs to the so-called four standard graph products, and it is the only one whose edges project to edges to both factors. On the other hand, direct product seems to be quite elusive from many perspectives. Let us mention connectivity, where it can occur that G × H is disconnected even when both factors G and H are connected. This happens when both G and H are bipartite as shown in [23], see also [10].
We start with Roman domination in direct products of complete graphs that yields a palette of sharp examples for the bounds that follows.

Proposition 2.1. For integers r and t we have
Proof. Let V (K r ) = {v 1 , . . . , v r } and V (K t ) = {u 1 , . . . , u t }. Let first t ≥ r = 2. In this case we have Then there is either one vertex in W 2 and one in W 1 ; or three vertices in W 1 and no in W 2 . The last option is not possible since K 2 ×K t contains at least four vertices. Also, the first possibility leads to a contradiction because there are at least two vertices nonadjacent to the unique vertex Suppose that γ R (K 3 × K t ) < 5 and let g(W 0 , W 1 , W 2 ) be a γ R (K 3 × K t )-function. Clearly, W 2 = ∅ because there are at least nine vertices in K 3 × K t . Every vertex is nonadjacent to exactly t + 1 ≥ 4 other vertices, and therefore |W 2 | = 2. For any pair of vertices (v i , u j ), (v k , u ℓ ) there exists at least one vertex nonadjacent to both. Indeed, if j = ℓ, then (v m , u j ) is such for {i, k, m} = {1, 2, 3} and if j = ℓ, then (v i , u ℓ ) and (v k , u j ) are such. In both cases we obtain a contradiction, and the equality γ R (K 3 × K t ) = 5 holds.
Finally, let t ≥ r > 3. It is easy to check that the function f and |W 2 | ≤ 2. Clearly, |W 2 | > 0 since K r × K t contains at least 16 vertices. If |W 2 | = 1, then we obtain a contradiction because there exist at least six vertices in V (K r ×K t ) being nonadjacent to the unique vertex in W 2 . If W 2 = {(v i , u j ), (v k , u ℓ )}, then there exist at least two vertices nonadjacent to both. More detailed, if i = k, then there are at least two more vertices (v i , u m ), (v i , u p ) such that |{j, ℓ, m, p}| = 4. A symmetrical argument works when j = ℓ. When i = k and j = ℓ, the vertices (v i , u ℓ ), (v k , u j ) are not adjacent to vertices in W 2 , a final contradiction. Hence, the equality γ R (K r × K t ) = 6 holds when t ≥ r > 3.
We next describe several lower and upper bounds for γ R (G × H) in terms of different parameters on the factors of G and H.

Theorem 2.2. For any graphs G and H with no isolated vertex,
By the symmetry of G × H, it is also satisfied that γ R (G × H) ≥ ρ(H)γ R (G), which completes the proof of the lower bound.
Next, we proceed to the upper bound. Let f ′ (V 0 , V 1 , V 2 ) be a γ tR (G)-function and D a γ(H)-set. Now, we define W ⊆ V (H) as a set of minimum cardinality among all total dominating sets W ′ such that D ⊆ W ′ . Notice that |W | ≤ 2|D|. We consider a function g(W 0 , ∅, W 2 ) on G × H as follows.
, then g(x, y) = 2, and g(x, y) = 0 otherwise. We claim that g is an RDF on G × H. Let (u, v) ∈ W 0 and distinguish the next two cases.
From the previous cases, we deduce that g is an RDF on G × H, as required. Therefore, By the symmetry of G × H, it is also satisfied that γ R (G × H) ≤ 2γ(G)γ tR (H), which completes the proof.
The upper bound of Theorem 2.2 is sharp for instance for K r × K t , t ≥ r > 3 by Proposition 2.1 and since γ(K r ) = γ(K t ) = 1 and γ tR (K r ) = γ tR (K t ) = 3. We are not aware of any example that is sharp for the lower bound of Theorem 2.2. Moreover, in most cases it seems that one could even add a factor two and the bound is still valid. However this is not true in all cases. In particular, Theorem 2.3. The following statements hold for any graph H with γ t (H) = γ(H).
Observe that g is an RDF on G × H by the same reasons as in the proof of Theorem 2.2. Hence, which completes the proof of (i). Finally, (ii) is an immediate consequence of (i).
We continue with another lower and upper bounds on γ R (G × H). Before this, we need to give some well-known results.
Theorem 2.4. The following statements hold for any graphs G and H with no isolated vertex.
Theorem 2.5. For any graphs G and H with no isolated vertex, Proof. By Theorem 2.4 (ii) and (iv) we deduce the lower bound. The upper bound γ R (G × H) ≤ 2γ t (G)γ t (H) holds by Theorem 2.4 (i) and (iii). Finally, to complete the proof of the upper bound we only need to combine Theorems 2.2 and 2.4 (i).
All the bounds of Theorem 2.5 are sharp. The lower bound is sharp for the graph K 2 × K 2 . By Theorem 2.5, we get γ R (K 2 × K 2 ) ≥ 4, and the equality holds by Proposition 2.1. For the first upper bound, γ R (G × H) ≤ 2γ t (G)γ t (H) we observe the family P 4 × P 4k . Here γ t (P 4k ) = 2k, and we have γ R (P 4 × P 4k ) ≤ 8k. The equality follows by [17,Theorem 5]. Finally, the second upper bound γ R (G × H) ≤ 6γ(G)γ(H) is sharp for the family K r × K t , t ≥ r > 3, by Proposition 2.1.
We now improve the upper bound γ R (G × H) ≤ 2γ t (G)γ t (H) from Theorem 2.5. For this, let G be a graph with no isolated vertex and D a γ t (G)-set. Let D ′ ⊆ D be a dominating set of G of minimum cardinality. Notice that D ′ is not necessarily a γ(G)-set. For instance, in The set D\D ′ is denoted by K G (D) and is called a kernel of D. By k(G) we denote the maximum possible cardinality among all kernels K G (D) from all γ t (G)-sets D and their dominating subsets D ′ . For instance, in Theorem 2.6. For any graphs G and H with no isolated vertex, Proof. Let D G be a γ t (G)-set together with D ′ G for which K G (D G ) has maximum cardinality k(G) and D H be a γ t (H)-set together with D ′ H for which K H (D H ) has maximum cardinality k(H). Let We will show that f (V 0 , ∅, V 2 ) is an RDF on G × H. Notice that every vertex (u, v) has a neighbor in D G × D H because D G and D H are total dominating sets of G and H, respectively.   Proof. Let D be a γ(G)-set and let W be a γ t (H)-set. We consider the function f . It is readily seen that f is an RDF on G × H.
The bound above is tight for instance if we consider the family of direct product graphs P 3 × K n with n ≥ 3. By Theorem 2.8 we get γ R (P 3 ×K n ) ≤ 5. The equality is obtained by the same arguments as in the proof of Proposition 2.1 for γ R (K 3 × K n ). Other sharp examples are γ R (K 3 × K n ) = 5, by Proposition 2.1, and γ R (P 4 × P 3k ) = 6k, by [17,Theorem 5].

Rooted product graphs
Given a nontrivial graph G and a nontrivial graph H with root v ∈ V (H), the rooted product graph G • v H is defined as the graph obtained from G and H by taking one copy of G and n(G) copies of H and identifying the i th -vertex of G with the vertex v in the i th -copy of H for every i ∈ {1, . . . , n(G)} [9]. Figure 1 shows an example of a rooted product graph. For every vertex x ∈ V (G), H x will denote the copy of H in G • v H containing x. The restriction of any function f : We recall two results given by Kuziak et al. [19], which will be useful later.
We continue this section with a useful lemma.
The following statements hold for any vertex x ∈ V (G): . Thus, the function g, defined by g(x) = max{f (x), 1} and g(u) = f (u) whenever u ∈ V (H x ) \ {x}, is an RDF on H x and with that also on H.
Hence ω(f x ) ≥ γ R (H) − 1 and the proof of (i) is completed.
then f x is an RDF on H x and, as above, we obtain a contradiction. So, Proof. For (i) let x ∈ B f . By Lemma 3.3 (ii) we obtain that f (x) = 0. This implies that the function which completes the proof of (i).
For (ii) observe that from any γ R (G)-function and any γ R (H − v)-function we can construct an RDF on G • v H of weight γ R (G) + n(G)γ R (H − v). By the equality given in (i), it follows that , which completes the proof.
Next, we state the three possible values that γ R (G • v H) can reach. We might recall that graphs G and H with γ R (G • v H) equal to each of these three values were already given in [19], although there was not proved that these are the only possible values γ R (G • v H) can reach. In this sense, with the next results we completely settle the problem of the Roman domination in rooted product graphs, already initiated in [19].

Theorem 3.5. Let G be a graph without isolated vertices and let H be a nontrivial graph. If v ∈ V (H) is the root, then
associated to f and differentiate the following cases.
, and by Theorem 3.2, we deduce that is maximum among all vertices in B f . By Lemma 3.3 (ii) we obtain that f (z) = 0 and f (S) ≤ 1. We analyze the next two subcases. Subcase 2.1. f (S) = 1. In this subcase there exists a vertex y ∈ S ∩ V 1 . On the way to a contradiction we define a function h on H z as follows: h(z) = 2, h(y) = 0 and h(u) = f z (u) whenever u ∈ V (H z ) \ {z, y}. It is easy to see that h is an RDF on H z of weight ω(h) = γ R (H z ), i.e., h is a γ R (H z )-function. Next, from h, f z and any γ(G)-set D, we define a function f ′ on G • v H as follows.
For every x ∈ D, the restriction of , which is a contradiction. Subcase 2.2. f (S) = 0. In this subcase, we first proceed to show that if y ∈ A f ∩ V 2 , then ω(f y ) > γ R (H). Suppose that there exists a vertex y ′ ∈ A f ∩ V 2 such that ω(f y ′ ) = γ R (H). Proceeding analogously as in the Subcase 2.1, from f y ′ , f z and any γ(G)-set D we can construct an RDF on G • v H of weight γ(G) + n(G)(γ R (H) − 1), which is a contradiction. Hence, for every y ∈ A f ∩ V 2 , it follows that ω(f y ) > γ R (H), as required.
We now observe that, as f (S) = 0, every vertex Next, we define a function h on G as follows.
In [19], the authors provided some sufficient conditions under which the three possible values of Roman domination number of rooted product graphs are achieved. In addition, we next give two examples in which we can observe that these expressions of γ R (G • v H) are realizable.
• If G is a graph with no isolated vertex and H is the graph shown in Figure 1, then • If G is a graph with no isolated vertex and H is the path P 4 (where the root v ∈ V (P 4 ) is a vertex of degree one), then We next proceed to characterize the graphs Theorem 3.6. Let G be a graph with no isolated vertex and H any nontrivial graph with root v ∈ V (H). The following statements are equivalent.
Proof. First, we assume that (i) holds, i.e., Thus, A f is a dominating set of G. Hence, So, we have equalities in the inequality chain above, and as a consequence, we deduce that every vertex y ∈ A f ∩ V 2 satisfies that ω(f x ) = γ R (H). Thus, as H x ∼ = H, there exists a γ R (H)-function g such that g(v) = 2.
On the other hand, we assume that γ R (H − v) = γ R (H) − 1 and that there exists a γ R (H)function g such that g(v) = 2. Let D be a γ(G)-set and g ′ be a γ R (H − v)-function. From D, g ′ and g, we define a function h on G • v H as follows. For every x ∈ D, the restriction of h to V (H x ) is induced by g. Moreover, if x ∈ V (G) \ D, then h(x) = 0 and the restriction of h to V (H x − x) is induced by g ′ . Observe that h is an RDF on G • v H, and so γ R (G • v H) ≤ ω(h) = |D|γ R (H) + |V (G) \ D|(γ R (H) − 1) = γ(G) + n(G)(γ R (H) − 1). Therefore, Theorem 3.5 leads to γ R (G • v H) = γ(G) + n(G)(γ R (H) − 1), which completes the proof. Now, we characterize the graphs G and H (and the root v ∈ V (H)) that satisfy the equality Theorem 3.7. Let G ∼ = ∪K 2 be a graph with no isolated vertex and H any nontrivial graph with root v ∈ V (H). The following statements are equivalent.
Finally, we characterize the graphs (and the root) reaching the equality γ R (G• v H) = n(G)γ R (H). Observe that, as shown in Theorem 3.5, there are three possible expressions for the Roman domination number of G• v H. Thus, for the case of the equality γ R (G• v H) = n(G)γ R (H), the corresponding characterization can be derived by eliminating the previous results (Theorems 3.6 and 3.7) from the family of all graphs G without isolated vertices and all nontrivial graphs H with root v ∈ V (H). Proof. First, we assume that γ R (G • v H) = n(G)γ R (H). By Lemma 3.1 we have that γ R (H − v) ≥ γ R (H) − 1. If γ R (H − v) = γ R (H) − 1, then by Theorems 3.6 and 3.7 we deduce that n(G)γ R (H) = γ R (G • v H) ≤ γ R (G) + n(G)(γ R (H) − 1), which is a contradiction as γ R (G) < n(G). Therefore, γ R (H − v) ≥ γ R (H), as desired.
Clearly, one might recall that the characterizations above rely on the following fact. It is necessary to know the relationship between γ R (H − v) and γ R (H) in the graphs H used in the rooted product, as well as, the existence of γ R (H)-functions with labels 1 or 2 in their roots. Thus, an interesting open problem that it is then of interest is as follows.
Open question: For a given graph H and a vertex v ∈ V (H), which is the relationship that exists between γ R (H − v) and γ R (H)?
On the other hand, as mentioned before, studies on Roman domination in Cartesian product graphs were published in [24,26]; and studies on the rooted product graphs were initiated in [19], and continued here in our work. In consequence, it would be of interest to continue these studies by considering the hierarchical product of graphs (see [2]), which is a subgraph of the Cartesian product as well as supergraph of the rooted product.