1. Introduction and Preliminary Results
In general, we use the standard notation of graph theory; see [
1]. We consider simple, undirected, connected graphs
, where
denotes the set of vertices and
the set of edges of the graph. The order of a graph refers to the number of its vertices, and its size corresponds to the number of its edges. We say that a vertex
is a neighbour of a vertex
in a graph
G if there is an edge
. The set of all neighbours of
is called a neighbourhood of the vertex
x and is denoted by
. We also denote
. Generally for
, the vertices of
that are adjacent to the vertices of
U are called neighbours of
U and their set is denoted by
. By
we mean the degree of the vertex
x in
G. A vertex of degree one is called a leaf, every neighbour of a leaf is called a support vertex. If a support vertex is adjacent to at least two leaves, it is said to be a strong support vertex and a weak support vertex otherwise. For a graph
G, by
we mean the set of support vertices of
G and by
the set of leaves of
G.
We use , , , , and to denote a path, a cycle, a complete graph, a bipartite graph, and an edgeless graph, respectively.
A subset is called a dominating set of graph G if every vertex outside this set has at least one neighbour in the set B. The parameter denotes the cardinality of the smallest dominating set in graph G. Similarly, a subset is an independent set of graph G if any two vertices in the set I are not adjacent to each other. An empty set and a set containing only one vertex are also independent. An independent set that is not a proper subset of any other independent set is called a maximal independent set. The independence number of a graph G, denoted by , is the maximal size of an independent set in G.
1.1. Background
The idea of domination appears in chessboard problems described in 1862 by De Jaenisch [
2] and it was formalised by C. Berge [
3] and then by O. Ore [
4] in 1962 who was the first who used the terms dominating set and dominating number. In 1977, the survey of E.J. Cockayne and S.T. Hedetniemi [
5] collected results related to domination and also used the notation
as the domination number. Since then, much has been written about dominating sets and dominating numbers. Over the following decades, the theory of domination was extended, and a huge development of this theory is still being observed. Dominating sets are studied not only from a purely theoretical point of view, but also interesting applications are considered. A review of the literature shows that dominating sets and domination parameters can be found in different fields, ranging from health care technology, through peer-to-peer messaging, location problems, and robotics, to crisis management and air/land/naval defence; see [
6]. For these reasons, the concept of dominating sets and their variants is relevant in modern graph theory. Dominating sets combined with the property of independence as an additional restriction can be used to model some mathematical objects, winning strategies in certain games played on graphs, or optimal sets of decisions. In the literature, we can find a number of variants and generalisations of dominating sets in graphs. In [
7], more than 70 types of dominating sets and domination parameters were described and new types of dominating sets still appear.
The study of dominating sets has also led to various specialised types of dominating sets, either by introducing additional restrictions or by generalising the classical domination concept. Some of these types are described in [
7,
8,
9]. F. Harary and T.W. Haynes in [
10,
11] introduced double domination, which generalises domination in graphs, and more generally, the concept of
k-tuple domination, which has been studied in [
12,
13] too. Let
k be a positive integer. The subset
is said to be a
k-tuple dominating set if
for every vertex
. The parameter
is equal to the minimum cardinality of a
k-tuple dominating in the graph G. For
, the
k-tuple domination coincides with the double domination, which was also studied by M. Blidia, M. Chellali, and T.W. Haynes [
14,
15,
16]. A set
is a double dominating set of
G if each vertex
is dominated by at least two vertices in
S. D.W. Bange, A.E. Barkauskas, and P.J. Slater [
17], and F. Harary and T.W. Haynes [
10] defined and studied the concept of an efficient doubly dominating set. M. Chellali, A. Khelladi, and F. Maffray also considered these sets in [
18], but they referred to them as exact doubly dominating sets. A set
is an exact dominating set of
G if each vertex
is dominated by exactly two vertices of
S.
1.2. Definitions of Multiply Domination and Proper 2-Domination
Another of the extensively researched types of dominating sets is the concept of multiple dominating sets, which was introduced by J.M. Fink and M.S. Jacobson in [
19]. For any integer
, a subset
is referred to as a
p-dominating set if every vertex outside of
S has at least
p neighbours within
S. For
, the
p-dominating set coincides with the classical dominating set. When
, we obtain the concept of 2-dominating sets, which has been studied in [
20,
21]. The parameter
represents the cardinality of the smallest
p-dominating set in the graph
G. We say that vertex
is called exactly
p-dominated if
.
As we mentioned, 2-dominating sets are intensively studied. However, for
the literature survey shows that
p-dominating sets are not a very popular topic. This follows from the fact that large
p vertices not belonging to
p-dominating sets must have large degrees, and this restriction limits classes of graphs with nontrivial (i.e., different from
)
p-dominating sets. Furthermore, a set
is termed a kernel if it is both independent and dominating. The concept of kernels in graph theory originated in the field of digraphs and was introduced by J. von Neumann and O. Morgerstern in the context of game theory [
22]. Over the years, this topic has been extensively studied for various purposes, including list coloring, perfectness, and location problems. C. Berge, a renowned mathematician, made significant contributions to the study of kernels in digraphs and applied them to solve various mathematical problems [
23,
24]. Numerous variants and generalisations of kernels have been proposed in the literature, with works such as [
23,
24,
25,
26,
27] exploring different aspects of this concept. For example, if we consider a set that is both independent and 2-dominating, we obtain a 2-dominating kernel. The concept of a 2-dominating kernel ((2-d)-kernel in short) was introduced by A. Włoch (see [
28]) and was intensively studied over the following years; see [
29,
30,
31,
32,
33]. In 2020, T. Haynes, S.T. Hedetniemi, and M.A. Henning in [
8] included a section written by A. Hansberg and L. Volkman that collected and classified results related to multiply domination. Contributing to this research, in this paper, we study a special subclass of 2-dominating sets.
From the definition of a
p-dominating set, it follows that for any graph
G,
. By definition, any
p-dominating set is also a
k-dominating set for
. Thus, any 3-dominating set is also a 2-dominating set. Therefore, it is interesting to study 2-dominating sets that are not 3-dominating. This motivates the introduction of proper 2-dominating sets. The idea of defining proper dominating sets of some types appeared, for example, in [
34,
35], in relation to secondary domination.
Definition 1. A subset is called a proper 2-dominating set if it is a 2-dominating set but not a 3-dominating set.
In other words, is a proper 2-dominating set if there exists a vertex outside the set D that has exactly two neighbours within the set D. Since the set is 3-dominating, it is not a proper 2-dominating set. To simplify the notation, we will write -dset to denote a proper 2-dominating set and by the cardinality of the smallest -dset.
For any graph
G, we have the inequality:
Directly from the Definition 1, we obtain the following lemma:
Lemma 1. Let be an integer. Every vertex of degree belongs to a -dominating set for .
The -dsets can be applied to a wide range of practical problems, such as crisis management. Let us consider a particular inhabited area where a plan should be developed to provide the inhabitants in the event of a critical situation, for example related to damage to critical infrastructure. It is important that each resident has access to help and that this access is further secured in the event of further damage. From the point of view of optimising the problem, it is preferable to plan location points so that the number of distribution points is minimal and that each resident has direct access to at least two of these points in the event of problems at one of the points. By modelling the situation with a graph, the solution to this problem of the location of such points is to find the minimum 2-dominating set of graphs. In turn, the minimum -dsets determine the weakest places where direct access to only two points is possible.
2. Main Results
As mentioned above, a graph does not always have a
-dset; as an example, we can consider a path
or the star
(see
Figure 1). In this section, we give a complete characterisation of graphs that have a
-dset.
Theorem 1. A connected graph G of order has a -dset D if and only if there exists a vertex v such that:
- 1.
, or
- 2.
and .
Proof. Let G be a graph with a -dset D. Let us assume, by contradiction, that every vertex and or .
If , then according to Lemma 1, it belongs to set D. Now, if , it must have at least three neighbours in D. Since D is a -dset, there exists a vertex in the set that has at most two neighbours in D, which is a contradiction.
In contrast, let us consider the following cases.
- 1.
Let us assume that v is a strong support vertex with exactly two leaves in its neighbourhood, and . Let , and , where:
- (i)
such that ;
- (ii)
such that ;
- (iii)
such that .
Let us consider the following cases:
- (a)
.
Let be the vertex with the smallest degree in the graph . Note that each vertex of has a degree of at least two, otherwise contains a leaf, which contradicts the assumption that v has exactly two leaves in its neighbourhood. Then, the set where is a -dset and the vertex is exactly 2-dominated.
- (b)
and .
Then, the set is a -dset and each vertex of is exactly 2-dominated.
- (c)
, , and .
Then, the set is a -dset and the vertex v is exactly 2-dominated.
- (d)
, , and .
Then, , the set is a -dset, and the vertex v is exactly 2-dominated by .
- 2.
Let us assume that v is a weak support vertex with one leaf in its neighbourhood. Let and , where sets , are the same as in case 1. Let us consider the cases:
- (a)
.
Similarly as in case 1(a), let be a vertex with the smallest degree in the graph . Note that every vertex in the set has a degree of at least two. Otherwise, we obtain a contradiction to the assumption that the vertex v has exactly one leaf in its neighbourhood. Then the set , where , is a -dset and the vertex is exactly 2-dominated.
- (b)
and .
Then, the set is a -dset and each vertex of is exactly 2-dominated.
- (c)
, , and .
Let . Then, the set is a -dset and the vertex v is exactly 2-dominated.
- (d)
, , and .
Then, the graph has an order two and does not have a -dset.
- 3.
Let us assume that . Let and , where sets , are the same as in case 1. Let us consider the cases:
- (a)
.
Similarly as in case 1(a), let be a vertex with the smallest degree in the graph . Note that every vertex in the set has a degree of at least two in G. Otherwise, we obtain a contradiction with the assumption that v is neither a leaf nor a support vertex. Then the set , where , is a -dset and the vertex is exactly 2-dominated.
- (b)
and .
Then, the set is a -dset and each vertex of is exactly 2-dominated.
- (c)
, , and .
Then, the set is a -dset, where the vertices . The existence of at least two such vertices is guaranteed by the assumption that v is neither a leaf nor a support vertex. Furthermore, the vertex v is exactly 2-dominated.
- (d)
, , and .
Then, the graph has an order one and does not have a -dset. □
Now, we give another complete characterisation of graphs that have a -dset using a generalised corona of graphs.
Definition 2. The generalised corona of a graph G and the sequence is the graph such that and
If
in the above definition, for
, then we obtain the definition of a corona of two graphs
G and
H introduced by R. Frucht and F. Harary in [
36].
Theorem 2. Let H be a graph. The graph G has a -dset if and only if , , .
Proof. The proof is analogous to the proof of Theorem 1. □
Figure 2 shows the corona
. By Lemma 1, every leaf belongs to a 2-dominating set. The set
is not the
-dset because every vertex that is not a leaf is at least 3-dominated. Moreover, since
is not
-dset, then this graph does not have a
-dset.
Moreover, from the above theorems we see that the graph G does not have a -dset if and only if every vertex is a support vertex and has at least three leaves in its neighbourhood or x is a leaf itself. Therefore, we immediately obtain the following corollaries.
Corollary 1. If , then G has a -dset.
Corollary 2. If a graph G has a vertex v such that , then G has a -dset.
From the above corollaries, we can obtain characterisations of well-known graph products with -dset, such as the Cartesian product, the tensor product, and the strong product. These graph products can be found in the literature under various names. To avoid confusion, we recall necessary definitions.
Let us recall the definitions of some graph products. Let G and H be two disjoint graphs.
Definition 3. The Cartesian product of two graphs G and H is the graph such that and .
Definition 4. The tensor product of the graphs G and H is the graph such that and . The tensor product is also called a direct product or a categorical product.
Definition 5. The strong product of the graphs G and H is the graph such that and .
Some other properties of these products are studied in [
37,
38,
39]. Note that these graph products have a symmetric structure.
Moreover, graph products play an important role in studying different properties and invariants in graphs. To describe classes of graphs with a given property it is easier to study graphs whose structure can be characterised in terms of smaller and simpler graphs, so many existing results come from the study of products of graphs. Operations on graphs also allow us to build families of graphs with a -dset.
The following results specify when the Cartesian product, the tensor product, and the strong product have a -dset.
Theorem 3. If G and H are nontrivial connected graphs, then the graph has a -dset.
Proof. In the Cartesian product of two graphs , we have . Since the graphs G and H are nontrivial by Corollary 1, the graph has a -dset. □
Theorem 4. If G and H are connected graphs of order and , then the graph has a -dset.
Proof. Since in the tensor product of two graphs , we have and and by Corollary 1, the graph has a -dset. □
Theorem 5. If G and H are nontrivial connected graphs, then the graph has a -dset.
Proof. Since for two nontrivial connected graphs, the graph has a -dset. □
Now we will show relations between and . The relationships between parameters of domination are one of the main directions of research in the theory of domination. Bounds for domination parameters can often be expressed in terms of other graph invariants. This means that they can be easily computed.
Theorem 6. Let G be a connected graph with a -dset. If , then .
Proof. If , it means that every -set of G is not a -set. In other words, there is a vertex of the -set that is not 3-dominated. Thus, every -set is also a -set. Therefore, , which completes the proof. □
The converse implication is not true. That is, the equality
does not necessarily imply that
. Consider the tree
T from
Figure 3. Then,
and these parameters are realised by the set
. Moreover, the smallest 3-dominating set is the set
, so
.
Theorem 7. Let G be a connected graph of order , which has a -dset and let . If there exists a -set in the graph G that is not independent, then .
Proof. Assume that G is a graph such that . Let D be a -set that is not independent. If there exists a vertex that is exactly 2-dominated, then D is a -set. Otherwise, every vertex is 3-dominated. Since D is not independent, there exist two adjacent vertices . Let . The existence of this vertex is guaranteed because G has at least three vertices. Then, or or . Without loss of generality, assume that . We will show that z does not belong to the set D. If , then the set is also a 2-dominating set, contradicting the assumption that D is a -set. This means that . As D is a 3-dominating set, every vertex is 3-dominated. Now, consider the set . Then, the vertex x has exactly two neighbours in the set , and every vertex is dominated by at least two vertices of . This implies that is a -set, which completes the proof. □
From the above theorem, we obtain the following corollary.
Corollary 3. Let G be a connected graph of order , which has a -dset and let . If then every -set in the graph G is independent.
Theorem 8. If G is a connected graph of order , which has a -dset, then Proof. The left inequality follows directly from the definition of a -dset. We will demonstrate that adding one vertex to the -set is sufficient to obtain a -dset. Therefore, we have the case where and all -sets are independent, and we let D be an arbitrary -set. Thus, there exists a vertex that is not a leaf. Otherwise, we would have a graph without a -dset, which is a contradiction. Every -set is independent. Otherwise, if there is a -set that is not independent, then it is also a -set. This is a contradiction to the assumption that every -set is independent. Since the -set is independent, . Furthermore, each neighbour of x is dominated by at least 3 vertices. As x is not a leaf, it has at least two neighbours, denoted as y and z. We form the set . Thus, every neighbour of x is dominated by at least two vertices of . Hence x is exactly 2-dominated by y and z. The set is a -dset, and , which concludes the proof. □
The following corollary presents the proper 2-domination number in standard classes of graphs. In each of these classes, the equality holds.
Corollary 4. Let be integers. Then,
- (i)
;
- (ii)
;
- (iii)
;
- (iv)
To illustrate the example of graph where the equality is met, let us consider the tree T such that and . Then, and these parameters are realised by the set . Moreover, the set is -dset and .
Now, we will show that in trees the value of the 2-proper domination number is related to the existence of a (2-d)-kernel. In [
20,
40], the following theorems have been proved.
Theorem 9. ([
20])
. For a tree T the following are equivalent:- 1.
.
- 2.
T has a unique -set that also is a unique -set.
Theorem 10. ([
40])
. A tree T has a --kernel if and only if . From the two aforementioned theorems it follows that a tree T has a (2-d)-kernel if and only if it possesses exactly one minimal 2-dominating set. This leads to the final result.
Theorem 11. Let T be a tree of order n, , which has a -dset. Then, Proof. Let T be a tree, which has a -dset. By Theorem 2 we obtain , , , where is an arbitrary tree of order n, . By Theorems 9 and 10 it follows that a tree T has (2-d)-kernel if and only if there exists a unique smallest 2-dominating set, which is also a maximum independent set. Let us consider the following cases:
- 1.
A tree T has (2-d)-kernel J. Let us consider the following subcases.
- (a)
If the set J is also 3-dominating, then there exists a vertex x that is not a leaf. This means that x has at least two neighbours identified as and . We create the set . Consequently, each neighbour of x is dominated by a minimum of two vertices of . Moreover, x is exactly 2-dominated by and . The set is a -dset, and . Hence .
- (b)
If the set J is not 3-dominating, then there exists a vertex , which is exactly 2-dominated. Thus, J is also a -dset. Therefore, .
- 2.
A tree T does not have (2-d)-kernel. Let us consider the following subcases:
- (a)
If , then by Theorem 6 it follows that .
- (b)
If , then by Theorem 9 every -set of a tree T is not independent. By Theorem 7 we obtain .
This completes the proof. □