Some New Perspectives on Global Domination in Graphs

Abstract

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.

1. 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 𝐺=(𝑉,𝐸) with |𝑉|=𝑛. The set 𝑆⊆𝑉 is called a dominating set if 𝑁[𝑆]=𝑉. A dominating set 𝑆 is called a minimal dominating set (MDS) if no proper subset 𝑆^′ of 𝑆 is a dominating set.

The minimum cardinality of a dominating set in 𝐺 is called the domination number of 𝐺 denoted by 𝛾(𝐺), and the corresponding dominating set is called a 𝛾-set of 𝐺.

The complement 𝐺 of 𝐺 is the graph with vertex set 𝑉 in which two vertices are adjacent in 𝐺 if and only if they are not neighbors in 𝐺.

A dominating set 𝑆 of 𝐺 is called a global dominating set if it is also a dominating set of 𝐺. The global domination number 𝛾_𝑔(𝐺) is the minimum cardinality of a global dominating set of 𝐺. 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 𝑊_𝑛 is defined to be the join 𝐶_𝑛−1+𝐾₁ where 𝑛≥4. The vertex corresponding to 𝐾₁ is known as apex vertex, and the vertices corresponding to cycle 𝐶_𝑛−1 are known as rim vertices.

Duplication of an edge 𝑒=𝑢𝑣 of a graph 𝐺 produces a new graph 𝐺₁ by adding an edge 𝑒^′=𝑢^′𝑣^′ such that 𝑁(𝑢)=𝑁(𝑢^′) and 𝑁(𝑣)=𝑁(𝑣^′).

The shadow graph 𝐷₂(𝐺) of a connected graph 𝐺 is constructed by taking two copies of 𝐺, say 𝐺^′ and 𝐺^′′. Join each vertex 𝑢^′ in 𝐺^′ to the neighbors of the corresponding vertex 𝑢^′′ in 𝐺^′′.

A vertex switching 𝐺_𝑣 of a graph 𝐺 is the graph obtained by taking a vertex 𝑣 of 𝐺, removing all the edges incident to 𝑣 and adding edges joining 𝑣 to every vertex not adjacent to 𝑣 in 𝐺.

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 𝐶_𝑛 and 𝑊_𝑛. Moreover, we obtain the domination number and the global domination number for the shadow graph of 𝑃_𝑛 and the graphs obtained by switching of a vertex in 𝐶_𝑛 as well as in 𝑊_𝑛.

2. Main Results

Theorem 1. If 𝐶^′_𝑛 is a graph obtained by duplication of an edge in 𝐶_𝑛(𝑛≠3,𝑛≠5) by an edge then every 𝛾-set of 𝐶^′_𝑛 is a global dominating set of 𝐶^′_𝑛 and 𝛾(𝐶^′_𝑛)=𝛾_𝑔(𝐶^′_𝑛)=⌈𝑛/3⌉. Furthermore, the global domination number remains invariant under the operation of duplication of an edge for 𝐶_𝑛.

Proof. Let 𝐶^′_𝑛 be a graph obtained by duplication of an edge in 𝐶_𝑛(𝑛≠3,𝑛≠5). Without loss of generality, let the edge 𝑒=𝑣₁𝑣₂ of 𝐶_𝑛 be duplicated by an edge 𝑒^′=𝑣^′₁𝑣^′₂.
Let 𝑆 be a 𝛾-set of 𝐶^′_𝑛. Then 𝑆 is a dominating set of 𝐶^′_𝑛. Now, any two distinct vertices 𝑣_𝑖 and 𝑣_𝑗 with 𝑑(𝑣_𝑖,𝑣_𝑗)≠2 in 𝐶^′_𝑛 are enough to dominate 𝐶^′_𝑛 because the vertices which are not in 𝑁[𝑣_𝑖] must belong to 𝑁[𝑣_𝑗] in 𝐶^′_𝑛. Since every 𝛾-set of 𝐶^′_𝑛 contains at least two such vertices 𝑣_𝑖 and 𝑣_𝑗, it is a dominating set of 𝐶^′_𝑛. Hence, every 𝛾-set of 𝐶^′_𝑛 is a dominating set of 𝐶^′_𝑛 as well as of 𝐶^′_𝑛. Consequently, every 𝛾-set of 𝐶^′_𝑛 is a global dominating set of 𝐶^′_𝑛.
Now, consider a 𝛾-set of 𝐶^′_𝑛: 𝑣𝑆=³,𝑣₆,𝑣₉,…,𝑣_3𝑗(),𝑣if𝑛≡0mod3₃,𝑣₆,𝑣₉,…,𝑣_3𝑗∪𝑣_𝑛()𝑛if𝑛≡1or2mod3for1≤𝑗≤3. (1)
Since 𝑑(𝑣₃)=3=𝑑(𝑣_𝑛)=Δ(𝐶^′_𝑛), 𝑆 must contain 𝑣₃ and 𝑣_𝑛 for minimum cardinality. Also, to retain the minimum cardinality of 𝑆, it must contain the vertices 𝑣_3𝑗(2≤𝑗≤⌊𝑛/3⌋). That is, |𝑆|≥⌈𝑛/3⌉. Now, 𝑆 being a 𝛾-set of 𝐶^′_𝑛 is a global dominating set of 𝐶^′_𝑛 with minimum cardinality ⌈𝑛/3⌉. This implies that 𝛾(𝐶^′_𝑛)=𝛾_𝑔(𝐶^′_𝑛)=⌈𝑛/3⌉. Also, as reported in Sampathkumar [1], 𝛾_𝑔(𝐶_𝑛)=⌈𝑛/3⌉(𝑛≠3,𝑛≠5). Hence, 𝛾_𝑔(𝐶_𝑛)=𝛾_𝑔(𝐶^′_𝑛)=⌈𝑛/3⌉(𝑛≠3,𝑛≠5). Thus, the global domination number remains invariant under the operation of duplication of an edge by an edge in 𝐶_𝑛.

Theorem 2. If 𝐺 is the graph obtained by duplicating each edge of 𝐶_𝑛 by an edge then every 𝛾-set of 𝐺 is a global dominating set of 𝐺 and 𝛾(𝐺)=𝛾_𝑔(𝐺)=𝑛.

Proof. Let 𝑆 be a 𝛾-set of 𝐺. Then 𝑆 is a dominating set of 𝐺. Now, any two adjacent vertices 𝑣_𝑖 and 𝑣_𝑗 in 𝐺 are enough to dominate 𝐺 because the vertices which are not in 𝑁[𝑣_𝑖] in 𝐺 must belong to 𝑁[𝑣_𝑗] in 𝐺. Since every 𝛾-set of 𝐺 contains at least two such vertices 𝑣_𝑖 and 𝑣_𝑗, every 𝛾-set of 𝐺 is a dominating set of 𝐺. Hence, every 𝛾-set of 𝐺 is a dominating set of 𝐺 as well as of 𝐺. This implies that every 𝛾-set of 𝐺 is a global dominating set of 𝐺.
Let 𝑣₁,𝑣₂,…,𝑣_𝑛 be the vertices of 𝐶_𝑛. Consider a 𝛾-set of 𝐺, 𝑆={𝑣₁,𝑣₂,…,𝑣_𝑛}. 𝑆 being a 𝛾-set of 𝐺 is a dominating set of 𝐺 with minimum cardinality. Moreover, since 𝑣₁,𝑣₂,…,𝑣_𝑛 are the vertices of maximum degree in 𝐺 and from the nature of the graph 𝐺, it is clear that 𝑆={𝑣₁,𝑣₂,…,𝑣_𝑛} is of minimum cardinality. Hence, 𝑆 is a 𝛾-set of 𝐺 with minimum cardinality 𝑛. Therefore, 𝛾(𝐺)=𝑛. Now, 𝑆 being a 𝛾-set of 𝐺 is a global dominating set of 𝐺 with minimum cardinality 𝑛 which implies that 𝛾(𝐺)=𝛾_𝑔(𝐺)=𝑛 as required.

The following Theorem 3 can be proved by the arguments analogous to the above Theorem 2.

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

Theorem 4. If 𝑊^′_𝑛 is a graph obtained by duplicating an edge of 𝑊_𝑛 by an edge then 𝛾_𝑔𝑊_𝑛=𝛾_𝑔𝑊^′_𝑛=4𝑖𝑓𝑛=43𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (2)
That is, the global domination number remains invariant under the operation of duplication of an edge in 𝑊_𝑛.

Proof. Let 𝑉(𝑊_𝑛)={𝑣₁,𝑣₂,…,𝑣_𝑛−1}∪{𝑐}, where 𝑐 is the apex vertex of 𝑊_𝑛.
It is easy to observe that 𝛾_𝑔𝑊_𝑛=4if𝑛=43otherwise. (3)
Case I (When a rim edge of 𝑊_𝑛 is duplicated by an edge). Without loss of generality, let the rim edge 𝑒=𝑣₁𝑣₂ of 𝑊_𝑛 be duplicated by an edge 𝑒^′=𝑣^′₁𝑣^′₂.
For 𝑛=4, since the vertex 𝑐 is adjacent to each vertex of 𝑊^′_𝑛, it must belong to any global dominating set of 𝑊^′_𝑛. Moreover, any two vertices are adjacent to third vertex other than 𝑐 in 𝑊^′_𝑛. Therefore, any global dominating set of 𝑊^′_𝑛 must contain at least four vertices including 𝑐 which implies that 𝛾_𝑔(𝑊^′_𝑛)=4.
For 𝑛>4, the vertex 𝑐 dominates 𝑊^′_𝑛 while the vertex 𝑐 and any two adjacent rim vertices of 𝑊_𝑛 are enough to dominate 𝑊^′_𝑛. Therefore, any global dominating set of 𝑊^′_𝑛 must contain at least three vertices of 𝑊^′_𝑛. This shows that 𝛾_𝑔(𝑊^′_𝑛)=3.
Thus, 𝛾_𝑔𝑊^′_𝑛=4if𝑛=43otherwise. (4)
Case II (When a spoke edge of 𝑊_𝑛 is duplicated by an edge). Without loss of generality, let the spoke edge 𝑒=𝑐𝑣₁ be duplicated by an edge 𝑒^′=𝑐^′𝑣^′₁.
For 𝑛=4, any two vertices in 𝑊^′_𝑛 are adjacent to the third vertex and also any three vertices are adjacent to the fourth vertex in 𝑊^′_𝑛. Therefore, any global dominating set of 𝑊^′_𝑛 must contain at least four vertices which implies that 𝛾_𝑔(𝑊^′_𝑛)=4.
For 𝑛>4, clearly any global dominating set of 𝑊^′_𝑛 must contain either 𝑐 or 𝑐^′ to achieve its minimum cardinality. Moreover, any two adjacent rim vertices of 𝑊_𝑛 and the vertex 𝑐 are enough to dominate 𝑊^′_𝑛 and they also dominate 𝑊^′_𝑛. This implies that 𝛾_𝑔(𝑊^′_𝑛)=3.
Thus, 𝛾_𝑔𝑊^′_𝑛=4if𝑛=43otherwise. (5)
Hence, we have proved that 𝛾_𝑔𝑊_𝑛=𝛾_𝑔𝑊^′_𝑛=4if𝑛=43otherwise. (6)
That is, the global domination number remains invariant under the operation of duplication of an edge in 𝑊_𝑛.

Theorem 5. Every 𝛾-set of 𝐷₂(𝑃_𝑛) is a global dominating set of 𝐷₂(𝑃_𝑛), and 𝛾𝐷₂𝑃_𝑛=𝛾_𝑔𝐷₂𝑃_𝑛=𝑛2()𝑛𝑖𝑓𝑛≡0mod42+1𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (7)

Proof. Consider two copies of 𝑃_𝑛. Let 𝑣₁,𝑣₂,…,𝑣_𝑛 be the vertices of the first copy of 𝑃_𝑛 and 𝑢₁,𝑢₂,…,𝑢_𝑛 the vertices of the second copy of 𝑃_𝑛.
If 𝑆 is a 𝛾-set of 𝐷₂(𝑃_𝑛) then 𝑆 is a dominating set of 𝐷₂(𝑃_𝑛). Now, any two adjacent vertices 𝑣_𝑖 and 𝑣_𝑗 of 𝐷₂(𝑃_𝑛) are enough to dominate 𝐷₂(𝑃_𝑛) because the vertices which are not in 𝑁[𝑣_𝑖] in 𝐷₂(𝑃_𝑛) must belong to 𝑁[𝑣_𝑗] in 𝐷₂(𝑃_𝑛). Since every 𝛾-set of 𝐺 contains at least two such vertices 𝑣_𝑖 and 𝑣_𝑗, every 𝛾-set of 𝐷₂(𝑃_𝑛) is a dominating set of 𝐷₂(𝑃_𝑛). Hence, every 𝛾-set of 𝐷₂(𝑃_𝑛) is a dominating set of 𝐷₂(𝑃_𝑛) as well as of 𝐷₂(𝑃_𝑛). This implies that every 𝛾-set of 𝐷₂(𝑃_𝑛) is a global dominating set of 𝐷₂(𝑃_𝑛).
Case I (𝑛=2,3). For 𝐷₂(𝑃₂) and 𝐷₂(𝑃₃), clearly 𝑆={𝑣₁,𝑣₂} and 𝑆={𝑣₂,𝑣₃} are 𝛾-sets as well as global dominating sets with minimum cardinality respectively. Therefore, 𝛾(𝐷₂(𝑃_𝑛))=𝛾_𝑔(𝐷₂(𝑃_𝑛))=2=⌊𝑛/2⌋+1 for 𝑛=2,3.
Case II (𝑛≥4). (i) For 𝑛≡0(mod4)(i.e., 𝑛=4𝑘,𝑘∈𝑁), consider a 𝛾-set 𝑆={𝑣_2+4𝑖,𝑣_3+4𝑖/0≤𝑖≤𝑘−1} where |𝑆|=𝑛/2.
(ii) For 𝑛≡1(mod4)(i.e., 𝑛=4𝑘+1,𝑘∈𝑁), consider a 𝛾-set 𝑆={𝑣_2+4𝑖,𝑣_3+4𝑖/0≤𝑖≤𝑘−1}∪{𝑣_𝑛−1} where |𝑆|=⌊𝑛/2⌋+1.
(iii) For 𝑛≡2or3(mod4)(i.e., 𝑛=4𝑘+2 or 𝑛=4𝑘+3,𝑘∈𝑁), consider a 𝛾-set 𝑆={𝑣_2+4𝑖,𝑣_3+4𝑖/0≤𝑖≤𝑘−1}∪{𝑣_𝑛−2,𝑣_𝑛−1} where |𝑆|=⌊𝑛/2⌋+1.
Now, 𝑆 being a 𝛾-set of 𝐷₂(𝑃_𝑛) is a global dominating set of 𝐷₂(𝑃_𝑛) with minimum cardinality implying that |𝑆|=𝛾(𝐷₂(𝑃_𝑛))=𝛾_𝑔(𝐷₂(𝑃_𝑛)).
Thus, we have proved that 𝛾𝐷₂𝑃_𝑛=𝛾_𝑔𝐷₂𝑃_𝑛=𝑛2()𝑛if𝑛≡0mod42+1otherwise. (8)

Theorem 6. If 𝐺_𝑣 is a graph obtained by switching of a vertex in cycle 𝐶_𝑛(𝑛≥8) then 𝛾(𝐺_𝑣)=𝛾_𝑔(𝐺_𝑣)=3.

Proof. Let 𝑣₁,𝑣₂,…,𝑣_𝑛 be the successive vertices of 𝐺=𝐶_𝑛 and 𝐺_𝑣 denotes the graph obtained by switching of a vertex 𝑣 of 𝐺. Without loss of generality, let the switched vertex be 𝑣₁.
Consider a set 𝑆={𝑣₁,𝑣₂,𝑣_𝑛}. Then 𝑆 is a dominating set of 𝐺_𝑣 as all the vertices except the pendant vertices, namely, 𝑣₂ and 𝑣_𝑛, are in 𝑁[𝑣₁] while 𝑣₂ and 𝑣_𝑛 are already in 𝑆. Moreover, the set 𝑆 is a minimal dominating set of 𝐺_𝑣 because for any 𝑣_𝑖∈𝑆, the set 𝑆−{𝑣_𝑖} does not dominate the vertex 𝑣_𝑖 of 𝐺_𝑣. Furthermore, the vertex 𝑣₁ dominates (𝑛−2) vertices of 𝐺_𝑣 and the remaining two vertices are pendant vertices. Therefore, at least three vertices are required to dominate 𝐺_𝑣, and hence |𝑆|≥3. Thus, 𝑆 being a minimal dominating set with minimum cardinality is a 𝛾-set of 𝐺_𝑣 which implies that 𝛾(𝐺_𝑣)=3.
Now, we claim that the pendant vertices in 𝐺_𝑣 are enough to dominate remaining vertices of 𝐺_𝑣. Since the vertex which is not in 𝑁[𝑣₂] in 𝐺_𝑣 must belong to 𝑁[𝑣_𝑛] in 𝐺_𝑣, any 𝑆⊆𝑉 containing 𝑣₂ and 𝑣_𝑛 will be a dominating set of 𝐺_𝑣. Thus, 𝑆 is a dominating set of 𝐺_𝑣 as well as of 𝐺_𝑣. This implies that 𝑆 is a global dominating set of 𝐺_𝑣.
Since 𝑆 is a 𝛾-set of 𝐺_𝑣 as above, it is of minimum cardinality. Therefore, 𝛾_𝑔(𝐺_𝑣)=3. Thus, 𝛾(𝐺_𝑣)=𝛾_𝑔(𝐺_𝑣)=3 as required.

Theorem 7. If 𝐺_𝑣 is a graph obtained by switching of a rim vertex in a wheel 𝑊_𝑛 then 𝛾_𝑔𝐺_𝑣=2𝑖𝑓𝑛=4,53𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (9)

Proof. Let 𝑣₁,𝑣₂,…,𝑣_𝑛−1 be the successive rim vertices of 𝐺=𝑊_𝑛 and 𝐺_𝑣 denotes the graph obtained by switching of the vertex 𝑣 of 𝐺. Without loss of generality, let the switched vertex be 𝑣₁. Let 𝑐 be the apex vertex of 𝑊_𝑛.
Case I (𝑛=4,5). It is easy to observe that 𝑆={𝑣₁,𝑣₃} is a global dominating set of 𝐺_𝑣 with minimum cardinality. Therefore, 𝛾_𝑔(𝐺_𝑣)=2 for 𝑛=4,5.
Case II (𝑛≥6). Consider a set 𝑆={𝑐,𝑣₃,𝑣₄}. Then, 𝑆 is a dominating set of 𝐺_𝑣 as all the vertices except the vertex 𝑣₁ are in 𝑁[𝑐] in 𝐺_𝑣 while 𝑣₁ is dominated by 𝑣₃. Now, the vertex 𝑣₃ dominates all the vertices of 𝐺_𝑣 except the vertices 𝑣₁,𝑣₂,𝑣₄, and 𝑐. But the vertices 𝑣₁ and 𝑣₂ are dominated by the vertices 𝑐 and 𝑣₄ in 𝐺_𝑣 respectively while the vertices 𝑣₄ and 𝑐 are already in 𝑆. Therefore, 𝑆 is a dominating set of 𝐺_𝑣. Hence, 𝑆 is a dominating set of 𝐺_𝑣 as well as of 𝐺_𝑣. That is, 𝑆 is a global dominating set of 𝐺_𝑣. Moreover, two vertices, namely, 𝑐 and either of the vertices from 𝑣₃ and 𝑣₄ are enough to dominate 𝐺_𝑣. Since any two vertices in 𝐺_𝑣 are adjacent to the third vertex in 𝐺_𝑣, at least three vertices are essential to dominate 𝐺_𝑣. This implies that 𝛾_𝑔(𝐺_𝑣)=3. Thus, we have proved that 𝛾_𝑔𝐺_𝑣=2if𝑛=4,53otherwise. (10)

Theorem 8. If 𝐺_𝑣 is the graph obtained by switching of the apex vertex in wheel 𝑊_𝑛 then 𝛾(𝐺_𝑣)=𝛾_𝑔(𝐺_𝑣)=⌈(𝑛−1)/3⌉+1.

Proof. Let 𝑣₁,𝑣₂,…,𝑣_𝑛−1 be the successive rim vertices of 𝐺=𝑊_𝑛 and let 𝐺_𝑣 denote the graph obtained by switching of the vertex 𝑣 of 𝐺. Let the switched vertex be the apex vertex 𝑐 of 𝑊_𝑛.
Because 𝑐 is adjacent to every other vertex in 𝐺, 𝑐 is not adjacent to any other vertex in 𝐺_𝑣. Therefore, any dominating set for 𝐺_𝑣 must contain 𝑐. This implies that any global dominating set of 𝐺_𝑣 must contain 𝑐.
Case I (𝑛=4,6). It is easy to observe that 𝑆={𝑐,𝑣₁} and 𝑆={𝑐,𝑣₁,𝑣₃} are 𝛾-sets as well as global dominating sets of 𝐺_𝑣 of 𝑊₄ and 𝑊₆ respectively with minimum cardinality.
Therefore, 𝛾𝐺_𝑣=𝛾_𝑔𝐺_𝑣=()2=4−13()+1for𝑛=4,3=6−13+1for𝑛=6. (11)
Thus, 𝛾(𝐺_𝑣)=𝛾_𝑔(𝐺_𝑣)=⌈(𝑛−1)/3⌉+1 for 𝑛=4,6.
Case II (𝑛≠4,6). The vertex 𝑐 will be the isolated vertex in 𝐺_𝑣. Hence, 𝐺_𝑣=𝐶_𝑛−1∪𝐾₁. Therefore, 𝛾(𝐺_𝑣)=𝛾(𝐶_𝑛−1)+1. But as reported in Sampathkumar [1], 𝛾(𝐶_𝑛−1)=⌈(𝑛−1)/3⌉. This implies that 𝛾(𝐺_𝑣)=⌈(𝑛−1)/3⌉+1.
Because every dominating set of 𝐺_𝑣 must contain 𝑐 and 𝑐 is enough to dominate 𝐺_𝑣, 𝛾(𝐺_𝑣)=𝛾_𝑔(𝐺_𝑣).
Hence, 𝛾(𝐺_𝑣)=𝛾_𝑔(𝐺_𝑣)=⌈(𝑛−1)/3⌉+1 for 𝑛≠4,6.
Thus, we have proved that 𝛾(𝐺_𝑣)=𝛾_𝑔(𝐺_𝑣)=⌈(𝑛−1)/3⌉+1 as required.

3. 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.

Acknowledgment

The authors are highly thankful to the anonymous referees for their kind comments and fruitful suggestions on the first draft of this paper.