-DOMINATION REVISITED

. Kammerling and Volkmann [ J. Korean Math. Soc. 46 (2009) 1309–1318] introduced the concept of Roman 𝑘 -domination in graphs. For a fixed positive integer 𝑘 , a function 𝑓 : 𝑉 ( 𝐺 ) → { 0 , 1 , 2 } is a Roman 𝑘 -dominating function on 𝐺 if every vertex valued 0 under 𝑓 is adjacent to at least 𝑘 vertices valued 2 under 𝑓 . In this paper, inspired by the concept of alliances in graphs, we revisit the concept of Roman 𝑘 -domination by not-fixing 𝑘 . We prove upper bounds for the new variant in cactus graphs and characterize cactus graph achieving equality for the given bound. We also present a probabilistic upper bound for this variant.


Introduction
For graph theory notation and terminology not given here we refer to [12], and for the probabilistic methods notation and terminology we refer to [1].We consider finite and simple graphs  with vertex set  =  () and edge set ().The number of vertices of  is called the order of  and is denoted by  = ().The open neighborhood of a vertex  ∈  is  () =   () = { ∈  |  ∈ } and the closed neighborhood of  is  [] =   [] =  () ∪ {}.The degree of a vertex , denoted by deg() (or deg  () to refer to ), is the cardinality of its open neighborhood.We denote by () and ∆(), the minimum and maximum degrees among all vertices of , respectively.A vertex of degree one is referred as a leaf and a vertex adjacent to a leaf is referred as a support vertex.A strong support vertex is a support vertex adjacent to at least two leaves, and a weak support vertex is a support vertex adjacent to precisely one leaf.For a subset  of vertices of , the subgraph of  induced by  is denoted by  [𝑆].A subset  of vertices is an independent set if [] has no edge.The independence number, () of , is the maximum cardinality of an independent set.A cactus graph is a connected graph in which any two cycles have at most one vertex in common.
Hedetniemi et al. [14] introduced the concept of alliance in graphs.This concept has been further considered by several other authors, see for example [13,14,19] .Favaron et al. [9] initiated the study of offensive alliance in graphs.A subset  of vertices of a graph  = (, ) is a global offensive alliance if for every  ∈  − , | [] ∩ | ≥ | [] − |.The minimum cardinality of a global offensive alliance of  is called the global offensive alliance number of , and is denoted by   ().A subset  of vertices of a graph  = (, ) is a global strong offensive alliance if for every  ∈  − , | [] ∩ | > | [] − |.The minimum cardinality of a global strong offensive alliance of  is called the global strong offensive alliance number of , and is denoted by  ̂︀  ().The concept of global offensive alliance in graphs was further studied in, for example, [10,11,18,20,21].
A function  :  −→ {0, 1, 2} having the property that for every vertex  ∈  with  () = 0, there exists a vertex  ∈  () with  () = 2, is called a Roman dominating function or just an RDF.The weight of an RDF  is the sum  ( ) = ∑︀ ∈  ().The minimum weight of an RDF on  is called the Roman domination number of  and is denoted by   ().For an RDF  in a graph , we denote by   (or    to refer to  ) the set of all vertices of  with label  under  .Thus an RDF  can be represented by a triple ( 0 ,  1 ,  2 ), and we can use the notation  = ( 0 ,  1 ,  2 ).The mathematical concept of Roman domination, was developed by Cockayne et al. [7].Many variations, generalizations and applications of Roman domination parameters have been studied, and to see the latest progress until 2020 see [3][4][5].
In this paper, inspired by the concept of alliance number, we are interested in those RDFs  = ( 0 ,  1 ,  2 ) such that every vertex  ∈  0 has as many neighbors in  2 as in  0 .We call such RDF as an Roman global offensive alliance-like dominating function or just RGOADF.Thus, an RGOADF on a graph  is a function  = ( 0 ,  1 ,  2 ) such that  is an RDF and for every vertex The weight of an RGOADF  is defined as expected.The minimum weight of an RGOADF on  is called the Roman global offensive alliance-like domination number of  and is denoted by   ().
If  = ( 0 ,  1 ,  2 ) is an RGOADF, then clearly  is an RDF.Furthermore, assigning 2 to any global offensive alliance  of  and 0 to vertices of  () − , we obtain an RGOADF.Thus we have the following.

Bounds
Chambers et al. [2] proved the following upper bound for the Roman domination number of a graph.

Theorem 2 ([2]
).For any connected graph  of order  ≥ 3,   () ≤ 4/5, with equality if and only if  is  5 or obtained from /5 5 by adding a connected subgraph on the set of centers of the components of /5 5 .
We note that it can be seen that if  =   ( ≥ 6) then   () =  − 1 ̸ ≤ 4/5.Thus the 4/5 bound given in the Theorem 2 is not valid for the Roman global offensive alliance-like domination number in general.In this section we prove that the bound of Theorem 2 is valid for the Roman global offensive alliance-like domination number in cactus graphs.We first prove our proposed bound for trees.The proof is similar to the proof of Theorem 2. Lemma 3. Let  be a tree of order at least three and   ( ) ≤ 4| ( )|/5.If  ′ is a tree obtained from  by joining a vertex of  to the center of a star of order at least three, then   ( ′ ) < 4| ( ′ )|/5.
Theorem 4. For any tree  of order  ≥ 3,   ( ) ≤ 4/5, with equality if and only if  ( ) can be partitioned into sets inducing  5 such that the subgraph induced by the central vertices of these paths is connected.
By Lemma 3, we may assume that deg( −1 ) = 2. Furthermore, we can deduce that deg( 1 ) = 2 by reversing the root at   .We consider the following cases.Let  1 be the number of children of  −3 that are support vertices of degree two,  2 be the number of children of  −3 of degree two having a child which is a support vertex of degree two, and  3 be the number of children of  −3 that are leaves.Furthermore, let  1 , . . .,  1 be the children of  −3 that are support vertices of degree two if  1 > 0,  1 , . . .,  2 be the children of  −3 of degree two having a child which is a support vertex of degree two if  2 > 0, and  1 , . . .,  3 be the children of Then we extend  ′ to an RGOADF  by assigning 2 to each support vertex of   −3 , 1 to  −3 and 0 to each other vertex of Assume that max{ 1 ,  2 } > 1.By the inductive hypothesis,   ( ′ ) ≤ 4 ′ /5.Let  ′ be a   ( ′ )-function.
Then we extend  ′ to an RGOADF  by assigning 2 to each support vertex of   −3 at distance two from  −3 , 2 to  −3 , 1 to each leaf of   −3 at distance two from  −3 and 0 to each other vertex of   −3 .Then    By the inductive hypothesis,  ( ′ ) can be partitioned into sets inducing  5 such that the subgraph induced by the central vertices of these paths is connected.Now  ( ) can be partitioned into sets inducing  5 such that the subgraph induced by the central vertices of these paths is connected.
We next prove the 4/5 bound of Theorem 4 for cactus graphs.We call a vertex  in a cycle  of a cactus graph  a special cut-vertex if  belongs to a shortest path from  to a cycle  ′ ̸ = .We call a cycle  in , a leaf-cycle if  contains at most one special cut-vertex.So the leaf cycle of a unicyclic graph contains no special cut vertex.A vertex  of a leaf-cycle  is a tree-root vertex if it has a neighbor  outside  such that the component of  −  containing  is a tree.Theorem 5.For a cactus graph  order  ≥ 3,   () ≤ 4/5, with equality if and only if  =  5 or  is a cactus graph in which  () can be partitioned into sets inducing  5 such that the subgraph induced by the central vertices of these paths is a connected cactus graph.
Proof.First note that if  is a cactus graph in which  () can be partitioned into sets inducing  5 such that the subgraph induced by the central vertices of these paths is a connected cactus graph, then by Theorem 2,   () = 4/5, and so   () ≥ 4/5 by Observation 1.On the other hand let  be a function that assigns 2 to the central vertex of each  5 , 0 to each vertex of degree two and 1 to every leaf.Then  has weight 4 on each  5 , and every vertex  with  () = 0 has degree precisely two and is adjacent to a vertex  with  () = 2. Thus,  is an RGOADF of weight 4/5 implying that   () ≤ 4/5.Consequently,   () = 4/5.In addition, it can be easily seen that   ( 5 ) = 4 = 4/5.
We prove by an induction on the order  of a cactus graph  that   () ≤ 4/5, and if equality holds then  =  5 or  () can be partitioned into sets inducing  5 such that the subgraph induced by the central vertices of these paths is a connected cactus graph.For the base step of the induction it is easy to see that the result holds if  ≤ 5. Thus assume that  ≥ 6. Assume the result holds for all cactus graphs of order 5 <  ′ < , and now consider the cactus graph  of order .The result is obvious by Theorem 4 if  is a tree.Thus assume that  has at least one cycle.Clearly,  contains a leaf-cycle.Let  be a leaf-cycle of .
Assume that  contains some tree-root vertex.Let  0 be a tree-root vertex of  such that deg( 0 ) is as maximum as possible, and  0 , . . .,   , where  ≥ 1, be a path from  0 to a farthest leaf   of  such that the component of  −  0  1 containing  1 is a tree.We consider the following three cases on .

Probabilistic bounds
Several upper bounds for the offensive alliance and global offensive alliance numbers are given by Rodr íguez-Velazquez et al. [18] and Harutyunyan [10,11].Harutyunyan [10] presented an important probabilistic upper bound for the global offensive alliance number of a graph, which was improved slightly by Jafari Rad [15], as follows.
Theorem 6 ( [15]).Let  = (, ) be a graph of order , maximum degree ∆ and minimum degree .For 1/2 >  > 0, With the same argument presented in Theorem 6, we present the following probabilistic upper bound for the Roman global offensive alliance-like domination number of a graph.Theorem 7. Let  = (, ) be a graph of order , maximum degree ∆ and minimum degree .For 1/2 >  > 0, Proof.We follow the proof of Theorem 6 given in [15].Create a subset  ⊆  by choosing each vertex  ∈  , independently, with probability  = 1/2 + .For every vertex  ∈  , let   denote the number of vertices in the neighborhood of  that are in .Let Thus the result follows.

Problems
It is clear that   () ≤  for any graph  of order  by assigning 1 to every vertex.In Theorem 5 we have shown that for every cactus graph  order  ≥ 3,   () ≤ 4/5.We note that the 4/5 bound is not valid even for graphs with minimum degree one.To see this, if  is obtained from   ( ≥ 11) by adding a leaf to a vertex, then   () =  − 2 > 4/5.Furthermore, the above example demonstrates that   () −   () can be arbitrarily large in a graph .Thus we propose the following problem.
Problem 1. Find best upper bounds for the Roman global offensive alliance-like domination number of a graph (or at least graphs with minimum degree at least two) in terms of the order.Problem 2. Does there is a polynomial algorithm that computes the Roman global offensive alliance-like domination number of a tree?