Hn-Domination in Graphs

The aim of this article is to introduce a new definition of domination number in graphs called hn-domination number denoted by . This paper presents some properties which show the concepts of connected and independent hn-domination. Furthermore, some bounds of these parameters are determined, specifically, the impact on hn-domination parameter is studied thoroughly in this paper when a graph is modified by deleting or adding a vertex or deleting an edge.


Introduction:
In this work a graph  = (, ) ( for simplicity) is a simple, finite and undirected graph.Every term which is not found here can be found in (1,2,3).
Let  and  represent vertex set and edge set respectively for graph .Consider a vertex  belongs to , the number of edges incident on a vertex  is called the degree of it and is denoted by () with minimum and maximum degree () and ∆(), respectively.In case ∆() = (),  is called regular graph.A subgraph  1 of a graph  is graph having all of its vertices and edges, a spanning subgraph is a subgraph has all vertices of .A set  ⊆  is an independent set or stable set in graph  if its vertices are not adjacent (4).Let G be a graph.The set  ⊆  is called dominating if each vertex belongs to  −  is adjacent to a vertex in .The minimum cardinality of all dominating sets is called the domination number of  and denoted by () (4).The first time that the concept of domination number of a graph appeared was in (5).In (6), the first survey published some result about this concept.Recently many papers have been written on domination in graphs like (7,8,9,10).Here, a new definition is introduced called hn-domination.Some fundamental results on hn-domination are presented.Further several bounds for the hndomination number are stated.Also, the effects on hn-domination parameter are presented when a graph is modified by deleting a vertex or deleting or adding an edge.

:
Definition 2.1: "Let G be a graph and D is a dominating set, the set D is called an hn-dominating set if for all adjacent u, v ∈ V − D there are two adjacent z 1 , z 2 ∈ D such that u is adjacent to z 1 and v is adjacent to z 2 (may be z 1 = z 2 )."Definition2.2:"Let G be a graph and D is hndominating set (hn-DS), then D is called minimal hn-dominating set (hn-MDS) if it has no proper hndominating set.(see Fig. 1).Definition2.3:"The minimum cardinality of a minimal hn-dominating set is called hn-domination number and denoted by γ hn (G).  .Case 2: If  is odd, then consider the set  1 = { 2+2 ,  = 0,1, … , ⌊  2 ⌋ − 1},  1 is a"dominating set"in the cycle of order .At the same time, the set  1 is not a ℎ −  in  since the two vertices   ,  1 are adjacent in  −  1 and  −1 and  2 are not adjacent in  1 .Thus, we must add either   or  1 to the set  1 to obtain ℎ − , say   .Therefore,  =  1 ∪ {  }.Again, In the same manner in Proposition 2.7,  is the minimum ℎ − , so  ℎ () = ⌈  2 ⌉.Thus, by the results of above two cases, we get the required result.
Observation 2.10: The domination number "for graphs   ,   , and

.
Proof: Let  be a  ℎ -set of a graph .To prove the lower bound; we take the two induced subgraphs 〈〉  〈 − 〉 to be null.The edges which can appear in this case are only the edges that joining between the vertices of  and  − .The minimum number of edges in this case can be determined when each vertex in  −  is dominated by only one vertex in .Therefore, the minimum number of edges in this case is | − |.Now, it is obvious that the upper bound occurs when a graph  is complete.Thus, the result is calculated.̅̅̅ ) = 2.The second way when  > 5, then we choose two vertices in   say  and  such that (, ) = 3.Thus, in   ̅̅̅ the vertex  is adjacent to all vertices in   ̅̅̅ except two vertices which are adjacent to it in   .Also, the vertex  is adjacent to these two vertices.Therefore,  and  belong to ℎ −  in   ̅̅̅ .Thus, in this case  ℎ (  ̅̅̅ ) = 2. 2) Since   ̅̅̅̅ ≅   ̅̅̅ ∪  1 , then by the same procedure in (1) and observation 2.10  ℎ (  ̅̅̅̅ ) =  ℎ (  ̅̅̅ ) + 1.
3) a) If  ≅  , , then the graph  ̅ contains two components; one of them is a complete graph of order  and the other is a complete graph of order .Thus, by using observation 2.10  ℎ ( , ̅̅̅̅̅̅ ) = 2.
b) If  ≅   , it is easy to check that  ℎ ( 2 ̅̅̅ ) =  ℎ ( 3 ̅̅̅ ) = 2. Now, there are three cases depending on the order of path as follows: i) If  = 4, then  4 is self complementary, then  ℎ ( 4 ̅ ) =  ℎ ( 4 ) = 2. ii) If  = 5, then the pendent vertices  and  become the two vertices which are dominating all vertices in  5 ̅̅̅ .Thus,  ℎ ( 5 ̅̅̅ ) = 2. iii)If  ≥ 6, then by the same manner in 1(iii), we get the result.4) It is obvious.□ Theorem 2.18.Let G be a graph has hn-domination number γ hn ,"then in G − v, v ∈ D if v is adjacent to at least two of the independent vertices"in V − D such that there is no vertex in D dominated these vertices, then γ hn (G − v) ≥ γ hn (G).Otherwise, γ hn (G − v) ≤ γ hn (G).Proof: "Let D be hn − MD with minimum cardinality of the graph G, then there are two cases as follows: Case 1: If we delete a vertex , where  ∈  then four cases are obtained as follows: i) if  is adjacent to at least two of the independent vertices"in  −  such that there is no vertex in  that dominate on these vertices, then these vertices must belong to  − .Thus,  ℎ ( − ) >  ℎ ().(forexample, see Fig. 2d).ii) If  is isolated in , then  ℎ ( − ) <  ℎ ().iii) If  is isolated in  and the neighborhoods of  in  −  are dominated by some vertices in the set , then  ℎ ( − ) <  ℎ ().(as an example, see Fig. 2b).iv) If  is the only vertex adjacent to  vertices in  −  and there is a vertex from the  vertices that dominates the other vertices, then in these cases  ℎ ( − ) =  ℎ ()( for example , see Fig. 2c,  = 1).Case 2: If we delete a vertex  from  − , then there are three cases as follows: i) If  ∈  is adjacent to  such that the neighborhoods of  in  −  are dominated by other vertex in  and  is not isolated in ,then  ℎ ( − ) <  ℎ () (for example, see Fig. 3b

Conclusion:
In this paper, we introduced a new definition for domination number in graphs, namely hn-domination.The hn-dominating set and hndomination number for some graphs are found and proved.Also, some operations in hn-domination number are stated and proved.Through this paper, we conclude some properties of hn-domination number.

Figure 2 .
Figure 2. Hn-domination number of a graph  −  when deletion a vertex from .

Figure 3 .
Figure 3. Hn-domination number of a graph  −  when deletion a vertex from  −

Figure 4 .Figure 5 .
Figure 4. Deletion an edge  that incident two vertices in  or in  −

Figure 6 .
Figure 6.Adding an edge for two vertices in  −

4 :
A set  is called  ℎ −set if it is hndominating set with cardinality  ℎ ().From the definition of spanning star there is a vertex such that all other vertices are adjacent with this vertex.Thus, the result is obtained.ii) By Definition 2.1, all isolated vertices must belong to any" ℎ − ."Therefore, if all other vertices in  that are not isolated are dominated by at least one vertex in  (in other words if there is a spanning star formed by the other vertices of ), then by using the previous step, we get the result.

Proposition 2.7: If
≅   , then  ℎ () = ⌊ In this case the lower bound in Proposition 2.11 does not change since  is an independent set.Since the two induced subgraphs 〈〉  〈 − 〉 can still be null graphs.The upper bound occurs Let  1 and  2 are adjacent in  − .Then two cases are obtained as follows: Case 1: if there is a vertex in  say  such that  1 and  2 are adjacent to , then  1 ,  2 and  makes a cycle.Case 2: if ∃ 1 ≠  2 ∈  such that  1 and  2 are adjacent to  1 and  2 respectively, then there is a cycle of order four for these vertices.If a graph ≅  1 ∪  2 ∪ … ∪   , then  ℎ () =  ℎ ( 1 ) +  ℎ ( 2 ) + ⋯ +  ℎ (  ).Proof: It is clear that every components in  has distinct hn-dominating set with hn-domination number  ℎ (  ),  = 1, … , .So  ℎ () =  ℎ ( 1 ) +  ℎ ( 2 ) + ⋯ +  ℎ (  ).A graph  can be classified into two classes depending on the value  where the graph is rregular as follows: Case 1: If  = 1, then the graph is complete of order two ( 2 ), so  ℎ (G) = 1.Case 2: If  ≥ 2, then the minimum domination number of  in this case is two, meaning that the graph is a cycle and by Proposition 2.7  ℎ (G) = ⌈ when 〈 − 〉 is complete.In this case all vertices in 〈 − 〉 must be adjacent to only one vertex in .Since, if two different vertices in  −  are adjacent to two different vertices in , then by the definition of ℎ −  , the two different vertices in  must be adjacent.Therefore, we obtain a contradiction with the hypothesis.Thus, the maximum number of edges found in the complete graph that contain the vertices of the set  −  with a vertex in .So, the required result is obtained.) In this case the upper bound in the Proposition 2.11 does not change, since a graph can be complete.The lower bound occurs when the induced subgraphs 〈〉 is a path.Since path is connected graph with minimum edges and size of path of order | − | is | − | − 1.Therefore, we get the result.Proposition 2.13: If  be a graph has hndomination," then for every two adjacent vertices  1 and  2 in  −  , there is a cycle containing  1 and  2 .Proof: Proof: By proposition 2.12 for every two adjacent vertices v 1 and v 2 in V − D , there is a cycle contains v 1 and v 2 , since  has no cycle, then all vertices in  −  is not adjacent .So  −  is independent.Proposition 2.15: