Unique response strong Roman dominating func- tions of graphs

Given a simple graphG = (V,E) with maximum degree ∆. Let (V0, V1, V2) be an ordered partition of V , where Vi = {v ∈ V : f(v) = i} for i = 0, 1 and V2 = {v ∈ V : f(v) ≥ 2}. A function f : V → {0, 1, . . . , d 2 e+1} is a strong Roman dominating function (StRDF) onG, if every v ∈ V0 has a neighbor w ∈ V2 and f(w) ≥ 1 + d12 |N(w)∩ V0|e. A function f : V → {0, 1, . . . , d ∆ 2 e+ 1} is a unique response strong Roman function (URStRF), if w ∈ V0, then |N(w) ∩ V2| ≤ 1 and w ∈ V1 ∪ V2 implies that |N(w) ∩ V2| = 0. A function f : V → {0, 1, . . . , d∆2 e + 1} is a unique response strong Roman dominating function (URStRDF) if it is both URStRF and StRDF. The unique response strong Roman domination number of G, denoted by uStR(G), is the minimum weight of a unique response strong Roman dominating function. In this paper we approach the problem of a Roman domination-type defensive strategy under multiple simultaneous attacks and begin with the study of several mathematical properties of this invariant. We obtain several bounds on such a parameter and give some realizability results for it. Moreover, for any tree T of order n ≥ 3 we prove the sharp bound uStR(T ) ≤ 8n 9 .


Introduction
The original study of Roman domination was motivated by the defense strategies of the Roman Empire during the reign of Emperor Constantine the Great, 274-337 A.D. Emperor Constantine had the requirement that an army or legion could be sent from its home to defend a neighboring location only if there was a second army which would stay and protect the home. Thus, there are two types of armies, stationary and mobile. Each vertex with no army must have a neighboring vertex with a mobile army. Stationary armies then dominate their own vertices, and a vertex with two armies is dominated by its stationary army, and its open neighborhood is dominated by the mobile armies. This part of history of the Roman Empire gave rise to the mathematical concept of Roman domination, as originally defined and discussed by Stewart [13] in 1999, and ReVelle and Rosing [11] in 2000. The defensive strategy of Roman domination is based on the fact that every place in which there is established a Roman legion (a label 1 in the Roman dominating function) is able to protect itself from external attacks; and that every unsecured (i.e., weak) place (a label 0) must have at least a stronger neighbor (a label 2). In that way, if an unsecured place is attacked, then the stronger neighbor can send it one of the two legions to defend it.
Two examples of Roman dominating functions are depicted in Figure 1. Although these two functions ( Figure 1) satisfy the conditions to be Roman dominating functions, they correspond to two very different real situations. The unique strong place 2 in Figure 1(b) must defend up to 8 unsecured locations from possible external attacks. However, in Figure 1(a), the task of defending the unsecured locations is divided between several strong locations. These observations have let us to pose the following question: how many weak locations/places can be defended by a strong location occupied by two legions? Taking into account that a strong place must leave one of its legions to defend itself, the situation depicted in Figure 1(b) does not seem to be an efficient defensive strategy: the Roman domination strategy fails against a multiple attack situation. If several simultaneous attacks to weak places occur, then a single stronger place will be not able to defend its neighbors efficiently. With this motivation in mind, in [1] Alvarez-Ruiz et al., introduced the concept of strong Roman dominating function. Then in other references such as [2,7,9,15] the properties of this parameter have been studied. Let Let E v be the set of all edges incident with a vertex v in G, that is, www.ejgta.org Unique response strong Roman dominating functions of graphs | Doost Ali Mojdeh et al.
The minimum and maximum degree of G are denoted by δ(G) = δ and ∆(G) = ∆. A star S n of order n ≥ 2 is the complete bipartite graph K 1, n−1 . We call the center of a star to be a vertex of maximum degree. For two vertices u and v in a connected graph G, the distance d(u, v) between u and v is the length of a shortest (u, v)-path in G. The maximum distance among all pairs of vertices of G is the diameter of G, which is denoted by diam(G). For notations and terminologies, are not herein, see [14]. For a real-valued function For more details on domination in graphs see [4], and for other domination parameters see [5,8].
A Roman dominating function (RDF) on a graph G is a function f : V → {0, 1, 2} such that every vertex u for which f (u) = 0 is adjacent to at least one vertex v with f (v) = 2. The weight of a Roman dominating function is the sum ω(f ) = v∈V f (v), and the minimum weight of an RDF of G is called the Roman domination number of G, denoted by γ R (G), For further, see [6,10].
From now on, if f : The minimum weight over all strong Roman dominating functions on G is called the strong Roman domination number of G, denoted by γ StR (G). An independent strong Roman dominating function (IStRDF) of G is an StRDF such that the set of all vertices assigned positive values is independent. The independent strong Roman domination number i StR (G) is the minimum weight of an IStRDF of G. an StRDF of minimum weight is called a γ StR (G)-function and likewise i StR (G)-function is defined. An example of an StRDF and an IStRDF can be seen on the graph in Figure 1 (b), by assigning a 5 to the vertex of maximum degree, a 1 to the vertex of degree 2 and a 0 to the remaining vertices.
In [12], Rubalcaba and Slater studied Roman domination influence of parameters in which the interest is in dominating each vertex exactly once. The authors [12] also introduced the concept of unique response Roman functions which we will adapt the definition for strong Roman functions as follows: A function f : V → {0, 1, . . . , ∆ 2 + 1} with the ordered partition , is a unique response strong Roman dominating function, or just URStRDF, if it is a unique response strong Roman function and a strong Roman dominating function. The unique response strong Roman domination number, denoted by u StR (G), is the minimum weight of a URStRDF of G.
It is worth mentioning that every graph has a unique response strong Roman dominating function since (∅, V (G), ∅) is such a function. Moreover, if f = (V 0 , V 1 , V 2 ) is a URStRDF on G, then V 2 is a 2-packing set. In Figure 2, the black shaded pebble represents a stationary army and the white shaded pebble represents a traveling army. It is easy to check that an attack on any weak vertex of the graph will have three traveling army responding to the attacks. www.ejgta.org

Preliminary results
In this section, we give some results on the unique response strong Roman domination number of graphs. Most of these results are straightforward and so we omit the proofs.
Observation 2.4. For n ≥ 3, u StR (K n ) = n−1 2 + 1 and for n ≥ m ≥ 1, u StR (K n,m ) = n 2 + m. It is known that γ R (P n ) = γ R (C n ) = 2n/3 . Clearly any RDF on paths and cycles is strong. Moreover, since paths and cycles have minimum RDF that are unique response, the following result then is immediate.
The next result shows that the difference between u StR (G) and γ StR (G) can be arbitrarily large.
Proof. Let k ≥ 2 be an integer and let G k be a double star in which every support vertex is of degree 2k + 1. It can be seen that u StR (G) = 3k + 2, while γ StR (G) = 2k + 2.

Bounds
We provide in this section some upper and lower bounds for the unique response strong Roman domination number of a graph G in terms of maximum degree, minimum degree, the domination number, the diameter and the order of G. Obviously, every graph of order n, u StR (G) ≤ n, with equality if and only if each component of G has order at most two. Our next result improves the previous upper bound.
Theorem 3.1. For any graph G of order n, u StR (G) ≤ n − ∆ 2 , and furthermore, this bound is sharp for all graphs of order n with ∆(G) = n − 1.
Proof. Let v be a vertex of maximum degree, and consider the URStRDF implying the desired bound. The sharpness of the upper bound may be seen for all graph G of order n with ∆(G) = n − 1.
Proof. By pervious Theorem the proof is clear.
Theorem 3.8. Let G be a connected graph with diam(G) ≥ 3, then Furthermore, this bound is sharp for paths P 3k+2 with k ≥ 0.
Proof. Let diam(G) = d = 3m + t for some integers m ≥ 1 and t ∈ {0, 1, 2}. Let P = y 0 y 1 . . . y d be a diametral path in G, and let f : V (P ) → {0, 1, 2} be a URStRDF defined on P by assigning a 2 to every vertex in V 2 = {y 0 , y 3 , . . . , y 3m }, a 0 to N (V 2 ) and a 1 to the remaining vertices of P. Note that V 2 is a 2-packing set of P as well as of G. Define now a function g : V → {0, 1, . . . , ∆ 2 + 1} by g(x) = f (x) for x ∈ V (P ) and g(x) = 1 otherwise. We also define a function h : . . , 3m}, h(x) = 0 for every x ∈ N (y i ) such that i ∈ {0, 3, . . . , 3m} and h(y) = g(y) for any remaining vertex y. Clearly, h is a URStRDF on G and we have For sharpness, let G be a path P 3k+2 with k ≥ 0. Then u StR (G) = γ StR (G) = 2n 3 = 2(3k+2) 3 = 2k + 2. On the other hand, we have n = 3k + 2, diam(G) = 3k + 1. Thus, n − diam(G)−1 Recall that a vertex v ∈ S is said to have a private neighbor with respect to the set S if there exists a vertex w ∈ N (v) ∩ (V − S) for which N (w) ∩ S = {v}. Let pn[v, S] denote the set of private neighbors of v with respect to the set S. Theorem 3.9. If G is a graph with ∆(G) ≥ 3, then Proof. Let G be a graph with ∆(G) ≥ 3. If i StR (G) ≥ u StR (G), then since i StR (G) ≥ ∆ 2 + 1, the inequality holds. Hence, we assume that i StR (G) < u StR (G), and let f = (V f 0 , V f 1 , V f 2 ) be an i StR (G)-function. Then there exist two vertices x 1 and y 1 ∈ V f 2 such that N (x 1 ) ∩ N (y 1 ) = ∅, for otherwise, by Observation 2.2, i StR (G) = u StR (G), a contradiction. Without loss of generality, we assume that d G (x 1 ) ≤ d G (y 1 ). It follows that the function f 1 defined by is an StRDF such that V f 1 2 is independent, where no edge of G joins V f 1 1 and V f 1 2 . By Observation 2.1, there is an IStRDF g 1 = (V g 1 0 , V g 1 1 , V g 1 2 ) of G with weight at most ω(f 1 ). Now using the facts that pn( Thereafter, if V g 1 2 is not a 2-packing, then there must exist two vertices x 2 and y 2 ∈ V g 1 2 with N (x 2 ) ∩ N (y 2 ) = ∅. As above, we can define a function f 2 and so on. Clearly, with this process we can get an IStRDF . Moreover, since This completes the proof.
The following corollaries are immediate consequences of Theorem 3.9 and Observation 2.2.
Corollary 3.2. If G is a graph with maximum degree three, then u StR (G) ≤ i StR (G).

URStRDF of trees
In this section, we show that for any tree T of order n ≥ 3, u StR (T ) ≤ 8n 9 and then we characterize some extremal trees which attain this upper bound. We now need to introduce some terminologies and notations. A vertex of degree one is called a leaf and its neighbor is called a support vertex. We denote the set of all leaves adjacent to support vertex v, by L v . For r, s ≥ 1, a double star S(r, s) is a tree with exactly two vertices that are not leaves, with one adjacent to r leaves and the other to s leaves. A rooted graph is a graph in which one vertex is labeled in a special way so as to distinguish it from other vertices. The special vertex is called the root of the graph.  Then ω(f ) = s+2r+4 2 ≤ 4 5 (s + r + 2) . A simple calculation shows that each condition of (i)-(iv) yields u StR (S(r, s)) = 4n 5 . Now suppose that for T = S(r, s), we have u StR (S(r, s)) = 4n 5 . We consider some cases. . Since k ≥ 4, and value 4 for s is not acceptable s−4+2k 10 ≥ 1. Therefore, for any value of s other than of value stated in the parts (i)-(iv) u StR (S(r, s)) < 4 5 (n) where n = r + s + 2. proof.
Following we show that this bound is sharp. Let G be a labeled graph on n vertices and let H be a rooted graph with root v. The rooted product graph G • v H is the graph obtained from G and n copies of H, say H 1 · · · H n , by identifying the root of the copy H i of H with the i th vertex of G, Godsil and McKay [3]. If H is a vertex transitive graph, then G • v H does not depend on the choice of v, up to isomorphism. In such a case we will just write G • H. Let S(K 1,4 )(the star K 1,4 with all its edges subdivided) be rooted in its center v and let F p m consist of all the rooted product graphs T o v S(K 1,4 ), where T is any tree on m ≥ 2 vertices (see Figure 3 for an example). Theorem 4.3. Let T be an n-vertex tree. If T ∈ F p m and m ≥ 2, then u StR (T ) = 8n 9 . Proof. Firstly, we notice that if the graph H = S(K 1,4 ) is an induced subgraph H of G, and its noncentral vertices have no neighbors outside H in G, then any URStRDF must put total weight at least 8 on the vertices of H. For the case of trees T ∈ F p m , m ≥ 2, they contain m disjoint induced subgraphs isomorphic to S(K 1,4 ) satisfying the situation mentioned above. So, u StR (T ) ≥ 16n 18 = 8n 9 for each T ∈ F p m , m ≥ 2. But since every tree T ∈ F p m has a vertex partition of m ≥ 2 sets including such subgraphs, a weight of at least 16 is needed on every set of such partition. Moreover, it is easy to find a u StR -function of weight 8n 9 , which leads to the equality.