Some New Perspectives on Global Domination in Graphs

A dominating set is called a global dominating set if it is a dominating set of a graph G and its complement G. Here we explore the possibility to relate the domination number of graph G and the global domination number of the larger graph obtained from G by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results.


Introduction
The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it.Many variants of dominating sets are available in the existing literature.This paper is focused on global domination in graphs.
We begin with simple, finite, and undirected graph  = (, ) with || = .The set  ⊆  is called a dominating set if [] = .A dominating set  is called a minimal dominating set (MDS) if no proper subset   of  is a dominating set.
The minimum cardinality of a dominating set in  is called the domination number of  denoted by (), and the corresponding dominating set is called a -set of .
The complement  of  is the graph with vertex set  in which two vertices are adjacent in  if and only if they are not neighbors in .
A dominating set  of  is called a global dominating set if it is also a dominating set of .The global domination number   () is the minimum cardinality of a global dominating set of .The concept of global domination in a graph was introduced by Sampathkumar [1].This concept is remained in focus of many researchers.For example, the global domination number of boolean function graph is discussed by Janakiraman et al. [2].The NP completeness of global domination problems is discussed by Carrington [3] and by Carrington and Brigham [4].The global domination number for the larger graphs obtained from the given graph is discussed by Vaidya and Pandit [5] while Kulli and Janakiram [6] have introduced the concept of total global dominating sets.The discussion on global domination in graphs of small diameters is carried out by Gangadharappa and Desai [7].
The wheel   is defined to be the join  −1 + 1 where  ≥ 4. The vertex corresponding to  1 is known as apex vertex, and the vertices corresponding to cycle  −1 are known as rim vertices.
Duplication of an edge  = V of a graph  produces a new graph  1 by adding an edge   =   V  such that () = (  ) and (V) = (V  ).
The shadow graph  2 () of a connected graph  is constructed by taking two copies of , say   and   .Join each vertex   in   to the neighbors of the corresponding vertex   in   .
A vertex switching  V of a graph  is the graph obtained by taking a vertex V of , removing all the edges incident to V and adding edges joining V to every vertex not adjacent to V in .
For the various graph theoretic notations and terminology we follow West [8] while the terms related to the concept of domination are used in the sense of Haynes et al. [9].
Here we consider the problem: Does the global domination number remain invariant under any graph operations?We present here an affirmative answer to this question for the graphs obtained by various graph operations on   and   .Moreover, we obtain the domination number and the global domination number for the shadow graph of   and the graphs obtained by switching of a vertex in   as well as in   .

Main Results
Since (V 3 ) = 3 = (V  ) = Δ(   ),  must contain V 3 and V  for minimum cardinality.Also, to retain the minimum cardinality of , it must contain the vertices V 3 (2 ≤  ≤ ⌊/3⌋).That is, || ≥ ⌈/3⌉.Now,  being a -set of    is a global dominating set of    with minimum cardinality ⌈/3⌉.This implies that (   ) =   (   ) = ⌈/3⌉.Also, as reported in Sampathkumar [1], ).Thus, the global domination number remains invariant under the operation of duplication of an edge by an edge in   .Theorem 2. If  is the graph obtained by duplicating each edge of   by an edge then every -set of  is a global dominating set of  and () =   () = .
Proof.Let  be a -set of .Then  is a dominating set of .Now, any two adjacent vertices V  and V  in  are enough to dominate  because the vertices which are not in [V  ] in  must belong to [V  ] in .Since every -set of  contains at least two such vertices V  and V  , every -set of  is a dominating set of .Hence, every -set of  is a dominating set of  as well as of .This implies that every -set of  is a global dominating set of .
Let V 1 , V 2 , . . ., V  be the vertices of   .Consider a -set of ,  = {V 1 , V 2 , . . ., V  }.  being a -set of  is a dominating set of  with minimum cardinality.Moreover, since V 1 , V 2 , . . ., V  are the vertices of maximum degree in  and from the nature of the graph , it is clear that  = {V 1 , V 2 , . . ., V  } is of minimum cardinality.Hence,  is a -set of  with minimum cardinality .Therefore, () = .Now,  being a -set of  is a global dominating set of  with minimum cardinality  which implies that () =   () =  as required.
The following Theorem 3 can be proved by the arguments analogous to the above Theorem 2.
That is, the global domination number remains invariant under the operation of duplication of an edge in   . Proof.
It is easy to observe that Case I (When a rim edge of   is duplicated by an edge).Without loss of generality, let the rim edge  = V 1 V 2 of   be duplicated by an edge Case II (When a spoke edge of   is duplicated by an edge).Without loss of generality, let the spoke edge  = V 1 be duplicated by an edge   =   V  1 .For  = 4, any two vertices in    are adjacent to the third vertex and also any three vertices are adjacent to the fourth vertex in    .Therefore, any global dominating set of    must contain at least four vertices which implies that   (   ) = 4.For  > 4, clearly any global dominating set of    must contain either  or   to achieve its minimum cardinality.
Moreover, any two adjacent rim vertices of   and the vertex  are enough to dominate    and they also dominate    .This implies that   (   ) = 3.Thus, Hence, we have proved that That is, the global domination number remains invariant under the operation of duplication of an edge in   .
Theorem 5. Every -set of  2 (  ) is a global dominating set of  2 (  ), and Proof.Consider two copies of   .Let V 1 , V 2 , . . ., V  be the vertices of the first copy of   and  1 ,  2 , . . .,   the vertices of the second copy of   .
If  is a -set of  2 (  ) then  is a dominating set of  2 (  ).Now, any two adjacent vertices V  and V  of  2 (  ) are enough to dominate  2 (  ) because the vertices which are not in Since every -set of  contains at least two such vertices V  and V  , every -set of  2 (  ) is a dominating set of  2 (  ).Hence, every -set of  2 (  ) is a dominating set of  2 (  ) as well as of  2 (  ).This implies that every -set of  2 (  ) is a global dominating set of  2 (  ).
(iii) For  ≡ 2 or 3(mod 4) (i.e.,  = 4 + 2 or Thus, we have proved that . ., V  be the successive vertices of  =   and  V denotes the graph obtained by switching of a vertex V of .Without loss of generality, let the switched vertex be Then  is a dominating set of  V as all the vertices except the pendant vertices, namely, V 2 and V  , are in [V 1 ] while V 2 and V  are already in .Moreover, the set  is a minimal dominating set of  V because for any V  ∈ , the set  − {V  } does not dominate the vertex V  of  V .Furthermore, the vertex V 1 dominates ( − 2) vertices of  V and the remaining two vertices are pendant vertices.Therefore, at least three vertices are required to dominate  V , and hence || ≥ 3. Thus,  being a minimal dominating set with minimum cardinality is a -set of  V which implies that ( V ) = 3.Now, we claim that the pendant vertices in  V are enough to dominate remaining vertices of  V .Since the vertex which is not in [V 2 ] in  V must belong to [V  ] in  V , any  ⊆  containing V 2 and V  will be a dominating set of  V .Thus,  is a dominating set of  V as well as of  V .This implies that  is a global dominating set of  V .
Since  is a -set of  V as above, it is of minimum cardinality.Therefore,   ( V ) = 3.Thus, ( V ) =   ( V ) = 3 as required.
Proof.Let V 1 , V 2 , . . ., V −1 be the successive rim vertices of  =   and  V denotes the graph obtained by switching of the vertex V of .Without loss of generality, let the switched vertex be V 1 .Let  be the apex vertex of   .
Case II ( ≥ 6).Consider a set  = {, V 3 , V 4 }.Then,  is a dominating set of  V as all the vertices except the vertex . ., V −1 be the successive rim vertices of  =   and let  V denote the graph obtained by switching of the vertex V of .Let the switched vertex be the apex vertex  of   .
Because  is adjacent to every other vertex in ,  is not adjacent to any other vertex in  V .Therefore, any dominating set for  V must contain .This implies that any global dominating set of  V must contain .
Because every dominating set of  V must contain  and  is enough to dominate  V , ( V ) =   ( V ).

Concluding Remarks
The concept of global domination is remarkable as it relates the dominating sets of a graph and its complement.We have explored this concept in the context of some graph operations and also have investigated the domination number and the global domination number for the larger graph obtained by some graph operations on a given graph.The invariance parameter for global domination is also explored.

Theorem 3 .
If  is the graph obtained by duplicating each edge of   by an edge then every -set of  is a global dominating set of  and () =   () = .Theorem 4. If    is a graph obtained by duplicating an edge of   by an edge then

Theorem 7 .
If  V is a graph obtained by switching of a rim vertex in a wheel   then  ( V ) = { 2   = 4,53 ℎ.
Without loss of generality, let the edge = V 1 V 2 of   be duplicated by an edge   = V  1 V  2 .Let  be a -set of    .Then  is a dominating set of    .Now, any two distinct vertices V  and V  with (V  , V  ) ̸ = 2 in    are enough to dominate    because the vertices which are not in [V  ] must belong to [V  ] in    .Since every -set of    contains at least two such vertices V  and V  , it is a dominating set of .Hence, every -set of    is a dominating set of    as well as of    .Consequently, every -set of    is a global dominating set of    .Now, consider a -set of    : since the vertex  is adjacent to each vertex of    , it must belong to any global dominating set of    .Moreover, any two vertices are adjacent to third vertex other than  in    .Therefore, any global dominating set of    must contain at least four vertices including  which implies that   (   ) = 4.For  > 4, the vertex  dominates    while the vertex  and any two adjacent rim vertices of   are enough to dominate    .Therefore, any global dominating set of is a global dominating set of  V .Moreover, two vertices, namely,  and either of the vertices from V 3 and V 4 are enough to dominate  V .Since any two vertices in  V are adjacent to the third vertex in  V , at least three vertices are essential to dominate  V .This implies that   ( V ) = 3.Thus, we have proved that If  V is the graph obtained by switching of the apex vertex in wheel   then ( and .But the vertices V 1 and V 2 are dominated by the vertices  and V 4 in  V respectively while the vertices V 4 and  are already in .Therefore,  is a dominating set of  V .Hence,  is a dominating set of  V as well as of  V .That is,