Restrained captive domination number

Zainab Yasir Alrikabi; Ahmed A. Omran; Hassan Jiad Al Hwaeer

doi:10.1515/eng-2022-0510

Article Open Access

Restrained captive domination number

Zainab Yasir Alrikabi , Ahmed A. Omran and Hassan Jiad Al Hwaeer

Published/Copyright: January 20, 2024

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From the journal Open Engineering Volume 14 Issue 1

Abstract

The restrained captive domination number (RCDN), denoted by γ Rca ( G ) , is a new definition of domination number in graphs introduced in this article. If D is a captive dominating set and G [ V − D ] has no isolated vertex, then D is a restrained captive dominating set (RCDS) of graph G . Furthermore, some properties, theorems, and propositions are determined. The inverse RCDN is introduced too. The RCDN in complement graphs is discussed. Finally, the RCDS of some graphs is determined. We have explained that if the graph has pendent vertex, then it has no RCDN.

Keywords: restrained captive domination number; restrained captive dominating set; inverse; complement

1 Introduction

In this work, a graph G is a simple, finite, and undirected graph, where V ( G ) = { v 1 , v 2 , … , v p } is the set of vertices and E ( G ) = { e 1 , e 2 , … , e q } is the set of edges, such that | V | = p and | E | = q , the number of edge incident on a vertex v 1 is called the degree of it, and it is denoted by deg ⁡ ( v 1 ) with minimum and maximum degree δ ( G ) and ∆ ( G ) , respectively [1 2 3]. Let G [ D ] be a subgraph of G induced by vertices in D . A subset D ⊂ V ( G ) is a captive dominating set if G [ D ] has no isolated vertex ( D is a total dominating set), and each vertex v ∈ D is adjacent to at least one vertex in V − D . The cardinality of a smallest captive dominating set of G , denoted by γ ca ( G ) , is the captive domination number of G . The symbol γ − 1 ( G ) represents the minimum cardinality over all inverse dominating set of G [4].

Therefore, many studies have been presented on and from these fields, topological graphs [5,6,7,8,9], number theory graph [10], general graphs [11,12,13,14,15,16], and others. The reader can found all notions that are not mentioned in previous studies [10,17,18]. The captive domination is initiated by Ahmed and colleagues [4], and they obtain many properties that are bounded. Later, Zainab and Ahmed determined the captive domination, complement of captive domination, and inverse of captive domination of many kinds of graphs [19]. In this work, the new properties are discussed, also for some graphs, and the restrained captive domination number (RCDN) and inverse RCDN (IRCDN) are determined for some graphs, and several bounds for RCDN are stated.

Figure 1

Relation between dominating set, CDS, and RCDS.

$Figure 2 Diagram of D D , D 1 {D}_{1} , and D 2 {D}_{2} .$

Figure 2

Diagram of D , D 1 , and D 2 .

Figure 3

G .

$Figure 4 C 6 {C}_{6} .$

Figure 4

C 6 .

$Figure 5 C 3 {C}_{3} .$

Figure 5

C 3 .

$Figure 6 F n {F}_{n} .$

Figure 6

F n .

$Figure 7 P n ⊙ P m {P}_{n}\odot {P}_{m} .$

Figure 7

P n ⊙ P m .

$Figure 8 G ≡ B n , n G\equiv {B}_{n,n} .$

Figure 8

G ≡ B n , n .

2 RCDN

Definition 2.1. Consider G be a graph, a subset D ⊂ V ( G ) is a restrained captive dominating set (RCDS) in G if D is a CDS and every vertex in V − D is adjacent to at least one vertex in V − D .

Let CDS be a captive dominating set and CDN be a captive domination number.

Definition 2.2. A subset D ⊂ V ( G ) is a minimal RCDS (MRCDS) if it has no proper RCDS.

Definition 2.3. The set D is a RCDN if:

D = min ⁡ { D i ; D i ∈ MRCDS } , and is denoted by γ Rca ( G ) . (Figures 1 and 2)

Example 2.4. In Figure 3.

D = { v , w } is the MCDS of G; thus, γ ca ( G ) = 2 .

D 1 = { u , w } is the MRCDS of G; thus, γ Rca ( G ) = 2 .

Observation 2.5: If G is of order n with RCDS and RCDN, then:

The order of G is n ≥ 4 .
deg ( v ) ≥ 2 , for all v ∈ V ( G ) .
δ ( G ) ≥ 2 , ∆ ( G ) ≥ 2 .
If v ∈ V − D , then N ( v ) ∩ D ≠ ∅ and N ( v ) ∩ V − D ≠ ∅ .
Induced subgraphs G [ D ] and G [ V − D ] have no isolated vertex.
γ ( G ) ≤ γ t ( G ) ≤ γ ca ( G ) ≤ γ Rca ( G ) .

Lemma 2.6

[3] If G is a graph in which the degree of each vertex is at least 2, then G contains a cycle.

Proposition 2.7

If a graph G has RCDN, then G contains a cycle. But the converse is not necessarily true.

Proof

Since every graph G has RCDN and the degree of each vertex is at least 2 by Observation 2.5. (2), then G contains a cycle by Lemma 2.6.

The converse is not necessarily true. For the example, take C 6 (Figure 4). Every vertex is in C 6 of degree 2 but C 6 has no RCDN.

Corollary 2.8

Any graph G of order n has a pendent vertex, and then, G has no RCDN.

Corollary 2.9

If G is a cycle graph, C n has RCDN, and n ≡ 0 ( mod 4 ) , then G [ V − D ] is an induced subgraph that has components of perfect matching graph.

Theorem 2.10

The RCDN of the K n , K n , m , W n , C n , and P n is as follows:

γ Rca ( K n ) = 2 , n ≥ 4 .
γ Rca ( K n , m ) = 2 , n , m ≥ 2 .
γ Rca ( W n ) = 2 .
γ Rca ( C n ) = n 2 if n ≡ 0 ( mod 4 ) .
γ Rca ( P n ) has no RCDN.

Proof

Since the set D is CDS by Observation 2.10. [4] and degree of every vertex in K n is n − 1 , this means that every vertex in V − D is adjacent to at least one vertex in V − D , and the result is obtained.
In a graph K n , m , there are two sets of vertices, namely, { v 1 , v 2 , … , v n } is first set and { u 1 , u 2 , … , u m } is second set in K n , m , such that every vertex in the first set is adjacent to all vertices in the second set. To create a RCDS, one of each set must belong to D , since the set D is CDS by Observation 2.10 [4] and every vertex in V − D is adjacent to at least one vertex in V − D , and the result is obtained.
Let D = { v , u } , where the vertex v is dominating vertex, since D is CDS by Observation 2.10. [4] and an induced subgraph in V − D is a path P n ; therefore, every vertex in V − D is adjacent to at least one vertex in V − D , and then, γ Rca ( W n ) = 2 .
If n ≡ 0 ( mod 4 ) , then it is axiomatic that the vertices of set D = v 2 + 4 i , v 3 + 4 i , i = 0,1 , … , n 4 − 1 represent the RCDS such that all adjacent vertices in the set D have maximum neighborhood and G [ V − D ] has no isolated vertex. So, D is the minimum RCDS. Thus, γ Rca ( C n ) = n 2 .
P n has no RCDN by Corollary 2.8.

Remark 2.11: If G has RCDN, then G has CDN, but the converse is not true.

Proof

By definition, the result is obtained.

The converse is not true. For the example, take C 3 (Figure 5). A graph C 3 has CDN equal 2 , but it has no RCDN.

Proposition 2.12

If G has CDN and it has at least two dominating vertices u and v one of them in V − D , then G has RCDN.

Proof

Suppose the vertex v ∈ V − D; since v is dominating vertex, which means it is adjacent to every vertex in V − D , then G has RCDN.

Proposition 2.13

If a graph G ≅ G 1 ∪ G 2 ∪ … ∪ G n , and a graph G has components G i , i = 1 , 2 , … , n , then γ Rca ( G ) = γ Rca ( G 1 ) + γ Rca ( G 2 ) + ⋯ + γ Rca ( G n ) .

Proof

It is clear that every component in G has distinct RCDS with RCDN ( G i ) , i = 1 , 2 , … , n . So, γ Rca ( G ) = γ Rca ( G 1 ) + γ Rca ( G 2 ) + ⋯ + γ Rca ( G n ) .

Observation 2.14: Consider G be a graph, with ∆ ( G ) = n − 1 . Then, G ̅ has no RCDS.

Remark 2.15: The graphs W n , ̅ S n , ̅ and K n ̅ have no RCDN.

Proof

Since W n ̅ , S n ̅ , and K n ̅ have no captive domination number by Remark 2.16 [4] and Observation 2.6 [4], then it has no RCDN.

Proposition 2.16

γ Rca ( C n ̅ ) = 2 if n ≥ 6 .

Proof

Since C n ̅ = E ( K n ) − E ( C n ) , this means that every vertex in C n ̅ has a regular degree equal to n − 3 , the set D is a CDS by Proposition 2.19 [4], and since C n ̅ [ V − D ] has no isolated vertex, then γ Rca ( C n ̅ ) = 2 .

Proposition 2.17

γ Rca ( P n ̅ ) = 2 , for n ≥ 6 .

Proof

Since P n ̅ = E ( K n ) − E ( P n ) , this means that there exist two vertices of degree n − 2 and n − 2 of vertices of degree n − 3 and the set D is a CDS by Proposition 2.18 [4], and since C n ̅ [ V − D ] has no isolated vertex, then γ Rca ( P n ̅ ) = 2 .

3 IRCDN

Definition 3.1: Consider G be a graph, and let D ⊂ V ( G ) be a MRCDN in G if V ( G ) − D , contains a RCDS D − 1 with respect to set D , then D − 1 is an inverse RCDS (IRCDS) in G , and the minimum cardinality of an IRCDS of G , denoted by γ Rca − 1 ( G ) , is called an IRCDN.

Proposition 3.2

γ Rca − 1 ( C n ) = n 2 , n ≥ 4 if and only if n ≡ 0 ( mod 4 ) .

Proof

If n ≡ 0 ( mod 4 ) , which is obvious by taking the remaining vertices, by Theorem 2.10 (4), then γ Rca − 1 ( C n ) = n 2 . Conversely, since γ Rca − 1 ( C n ) = n 2 , this means n ≡ 0,2 ( mod 4 ) , so C n has no RCDN in this modulus ( n ≡ 2 ( mod 4 ) ) . Therefore, n ≡ 0 ( mod 4 ) .

Observation 3.3

γ Rca − 1 ( K n ) = 2 , for n ≥ 4 .
γ Rca − 1 ( K n , m ) = 2 , for n , m ≥ 2 .

Proposition 3.4

γ Rca − 1 ( W n ) = 2 n 4 if n ≡ 0 , 3 ( mod 4 ) 2 n 4 − 1 if n ≡ 2 ( mod 4 ) 2 n 4 − 2 if n ≡ 1 ( mod 4 ) .

Proof

Let v 1 , v 2 , … … , v n be the vertices of graph W n and D ⊂ V ( G ) such that:

D − 1 = { v 2 + 4 i , v 3 + 4 i , i = 0 , 1 , . . . , n 4 − 1 } if n ≡ 0 ( mod 4 ) { v 2 + 4 i , v 3 + 4 i , i = 0 , 1 , . . . , n 4 − 2 } if n ≡ 1 ( mod 4 ) { v 2 + 4 i , v 3 + 4 i , i = 0 , 1 , … . , n 4 − 2 } ∪ { v n − 2 } if n ≡ 2 ( mod 4 ) { v 2 + 4 i , v 3 + 4 i , i = 0 , 1 , … , n 4 − 2 } ∪ { v n − 1 , v n − 2 } if n ≡ 3 ( mod 4 ) .

The maximum number of vertices that can be restrained captive dominated by two vertices is four. So, we can choose the middle vertices from any four sequent vertices. Then, there are four cases, which are as follows:

Case 1: If n ≡ 0 ( mod 4 ) , let D 1 − 1 = { v 2 + 4 i , v 3 + 4 i , i = 0,1 , … . . , n 4 − 2 } be IRCDS; all adjacent vertices in this set have maximum neighborhood. So, D 1 − 1 is the minimum RCDS and γ Rca − 1 ( W n ) = 2 n 4 .

Case 2: If n ≡ 1 ( mod 4 ) , let D 2 − 1 = { v 2 + 4 i , v 3 + 4 i , i = 0 , 1 , … . . , n 4 − 2 } , then it is clear that by the same way in Case 1, the set D 2 − 1 is the minimum RCDS. Thus, γ Rca − 1 ( W n ) = 2 n 4 − 1 .

Case 3: If n ≡ 2 ( mod 4 ) , let D 3 − 1 = { v 2 + 4 i , v 3 + 4 i , i = 0 , 1 , … . . , n 4 − 2 } , then by the same way in Case 1 and Case 2, D 3 − 1 is the minimum RCDS to vertices { v n , v 1 , … , v n − 2 } . So the remaining vertex v n − 1 is not dominated by any vertex in D 3 − 1 ; therefore, we need take a vertex v n − 2 in D 3 − 1 ; then, D − 1 = D 3 − 1 ∪ { v n − 2 } . Thus, γ Rca − 1 ( W n ) = 2 n 4 − 2 .

Case 4: If n ≡ 3 ( mod 4 ) , let D 4 − 1 = { v 2 + 4 i , v 3 + 4 i , i = 0 , 1 , … . . , n 4 − 2 } , then D 4 − 1 is MRCDS to vertices { v 1 , v 2 , … , v n − 3 } , so the remaining vertices in W n that are not dominated by D 4 − 1 are { v n − 1 , v n − 2 } ; then, D − 1 = D 4 − 1 ∪ { v n − 1 , v n − 2 } . Thus , γ Rca − 1 ( W n ) = 2 n 4 .

From all the aforementioned cases, we obtain the results.

4 RCDN and IRCDN of some graphs and its complements

Proposition 4.1

If G be a fan graph denoted by G ≡ F n ≡ P 1 ⊙ P m , then γ Rca ( G ) = 2 .

Proof

Consider G be a fan graph of order 1 + m such that 1 be the order of a path P 1 and m be the order of a path P m . The vertex set of the first path is { v 1 } and the vertex set of the second path is { u 1 , u 2 , … … , u m } ; since the vertex v 1 is dominating vertex, which means that dominates on all vertices in G , then D = { v 1 , u 1 } be minimum and RCDS, so γ Rca ( G ) = 2 .

Proposition 4.2

If G be a fan graph denoted by G ≡ F n ≡ P 1 ⊙ P m , then

γ Rca − 1 ( G ) = 2 m 4 if m ≡ 0,2,3 ( mod 4 ) 2 m 4 − 1 if m ≡ 1 ( mod 4 ) , and the graph G has no inverse RCDS when m = 1 , 2 .

Proof

Let G be a fan graph of order 1 + m such that P 1 be a path of order 1 and P m be a path of order m (as shown in the Figure 6), then depending on m , there are four cases as follows:

Case 1. If m = 1 , then the graph P 1 ⊙ P 1 ≡ K 2 , so one can be concluded that there is no CDS; therefore, there is no RCDS in this case.

Case 2. If m = 2 , then the graph P 1 ⊙ P 3 ≡ K 3 , so one can be concluded that there is no RCDS; therefore, there is no IRCDS in this case.

Case 3. If m ≡ 0,2,3 ( mod 4 ) , then γ Rca − 1 ( G ) ≡ γ Rca − 1 ( F n ) = γ ca ( P m ) = 2 m 4 by Theorem 2.12 [4].

Case 4. If m ≡ 1 ( mod 4 ) , then γ Rca − 1 ( G ) ≡ γ Rca − 1 ( F n ) = γ ca ( P n ) − 1 = 2 m 4 − 1 .

From all the aforementioned cases, we obtain the results.

Proposition 4.3

If G be a graph denoted by G ≡ F n ̅ ≡ P 1 ⊙ P m , ̅ then G has no RCDN.

Proof

Since a graph G has isolated vertex, then G has no RCDN by Observation 2.5 (2) and by Definition 2.1.

Proposition 4.4

If G be a corona graph denoted by G ≡ P n ⊙ P m , then

γ Rca ( G ) = 2 if n = 1 , m ≥ 3 n if n ≥ 2 , m ≥ 2 , the graph G has no RCDS when n = 1 , m = 1 , 2 and n ≥ 2 , m = 1 .

Proof

Consider G be a graph of order n + nm such that P n is a path of order n and P m is a path of order m , then depending on n and m , there are five cases as follows:

Case 1. If n = 1 , m = 1 then the graph P n ⊙ P m ≡ K 2 , so one can be concluded that there is no RCDS in this case.

Case 2. If n = 1 , m = 2 , then the graph P n ⊙ P m ≡ K 3 , so one can be concluded that there is no RCDS in this case.

Case 3. If n ≥ 2 , m = 1 , then G [ V − D ] has isolated vertices, so one can be concluded that there is no RCDS in this case by Observation 2.5 (5).

Case 4. If n = 1 , m ≥ 3 , then the graph P n ⊙ P m ≡ F n , then γ Rca ( G ) = 2 .

Case 5. If n ≥ 2 , m ≥ 2 , then let v 1 , v 2 , … … , v n be the vertices of path P n and let u 1 , u 2 , … … , u m be the vertices of path P m ; since every vertex in P n joins with vertices of copy path p m from u 1 to u m and G [ V − D ] has no isolated vertices, then the minimum dominating set D = { v 1 , v 2 , … … , v n } is RCDS and γ Rca ( G ) = n .

For all the aforementioned cases, we obtain the results.

Proposition 4.5

If G be a corona graph denoted by G ≡ p n ⊙ p m , then

γ − 1 Rca ( G ) = n 2 m 4 if n ≥ 1 , m ≡ 0 , 2 , 3 ( mod 4 ) n 2 m 4 − 1 if n ≥ 1 , m ≡ 1 ( mod 4 ) , and the graph G has no IRCDS when n = 1 , m = 1 , 2 and when n ≥ 2 , m = 1 .

Proof

Let G be a graph of order n + nm such that P n is a path of order n and P m is a path of order m (as shown in Figure 7) then depending on n and m then there are five cases as follows:

Case 1: If n = 1 , m = 1 , then the graph P 1 ⊙ P 1 ≡ K 2 , so one can be concluded that there is no inverse RCDS in this case.

Case 2: If n = 1 , m = 2 , then the graph P 1 ⊙ P 2 ≡ K 3 , so one can be concluded that there is no IRCDS in this case.

Case 3: If n ≥ 2 , m = 1 , then the graph G has pendent vertices, so there is no RCDS; therefore, there is no IRCDN in this case by Corollary 2.8.

Case 4: If n ≥ 1 , m ≡ 0 , 2 , 3 ( mod 4 ) , then there are two subcases as follows:

If n = 1 , m ≡ 0,2,3 ( mod 4 ) , then P 1 ⊙ P m ≡ F n , so γ Rca − 1 ( G ) = 2 m 4 by Proposition 4.2.
If n > 1 , m ≡ 0 , 2 , 3 ( mod 4 ) , then since there are n copies of F n connected by a path and every copy of F n , γ Rca − 1 ( F n ) = 2 m 4 by Proposition 4.2, then γ Rca − 1 ( G ) = n 2 m 4 .

Case 5: If n ≥ 1 , m ≡ 1 ( mod 4 ) , then there are two subcases as follows:

If n = 1 , m ≡ 1 ( mod 4 ) , then P 1 ⊙ P m ≡ F n , so γ Rca − 1 ( G ) = 2 m 4 − 1 by Proposition 4.2.
If n > 1 , m ≡ 1 ( mod 4 ) , by the same way in Case 4, then γ Rca − 1 ( G ) = n 2 m 4 − 1 .

From all the aforementioned cases, we obtain the results.

Proposition 4.11

If G ≡ B n , n be a barbell graph of order 2 n , then γ Rca ( G ) = 2 .

Proof

The vertex set of B n , n is { v i :1 ≤ i ≤ 2 n }, the vertex set of the first complete graph K n is { v 1 , v 2 , … … , v n } , and the vertex set of the second complete graph K n is { v n + 1 , v n + 2 , … … , v 2 n } as shown in Figure 8.

From the definition of a barbell graph, there are two copies of a complete graph, each of which can be dominated by one vertex, and to keep the totality of a dominating set, two adjacent vertices are taken. Thus, the set D = { v n , v n + 1 } is a minimum RCDS, and γ Rca ( G ) = 2 .

Proposition 4.12

If G ≡ B n , n , n > 2 be the barbell graph of order 2n, then γ Rca − 1 ( G ) = 4 .

Proof

By proposition 4.11, let = { v n , v n + 1 } , so it is obvious that the set D is restrained captive dominating and it is the minimum, and since there exists another dominating set in V − D and it is minimum, let S = { v 1 , v 2 } ∪ { v n + 2 , v n + 3 } , this set is inverse RCDN, then γ Rca − 1 ( G ) = 4 .

5 Conclusion

New definitions, properties, and boundaries of a new parameter have been discovered along the process of arriving at the results described in this article. Specially identified above are several graphs’ inverse and complement.

Conflict of interest: Authors state no conflict of interest.
Data availability statement: The most datasets generated and/or analysed in this study are comprised in this submitted manuscript. The other datasets are available on reasonable request from the corresponding author with the attached information.

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Received: 2023-05-19

Revised: 2023-08-02

Accepted: 2023-08-08

Published Online: 2024-01-20

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Keywords for this article

restrained captive domination number; restrained captive dominating set; inverse; complement

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