On ( 2- d ) -Kernels in Two Generalizations of the Petersen Graph

: A subset J is a ( 2- d ) -kernel of a graph if J is independent and 2-dominating simultaneously. In this paper, we consider two different generalizations of the Petersen graph and we give complete characterizations of these graphs which have ( 2- d ) -kernel. Moreover, we determine the number of ( 2- d ) -kernels of these graphs as well as their lower and upper kernel number. The property that each of the considered generalizations of the Petersen graph has a symmetric structure is useful in ﬁnding ( 2- d ) -kernels in these graphs.


Introduction
In general, we use the standard terminology and notation of graph theory (see [1]). Let G be an undirected, connected, and simple graph with the vertex set V(G) and the edge set E(G). The order of the graph G is the number of vertices in G. The size of the graph G is its number of edges. By P n , n ≥ 1 and C n , n ≥ 3, we mean a path and a cycle of order n, respectively.
Let G = (V, E) and G = (V , E ) be two graphs. If V ⊆ V and E ⊆ E, then G is a subgraph of G, written as G ⊆ G. If G ⊆ G and G contain all the edges xy ∈ E with x, y ∈ V , then G is an induced subgraph of G and we write G := V G . Graphs G and G are called isomorphic, and denoted by G ∼ = G , if there exists a bijection φ : V → V with xy ∈ E ⇔ φ(x)φ(y) ∈ E for all x, y ∈ V. The complement of the graph G is a graph G such that V(G) = V(G) and two distinct vertices of G are adjacent if and only if they are not adjacent in G. A graph G is called bipartite if V(G) admits a partition into two classes such that every edge has its ends in different classes.
A subset D ⊆ V(G) is a dominating set of G if each vertex of G not belonging to D is adjacent to at least one vertex of D. A subset S ⊆ V(G) is called an independent set of G if no two vertices of S are adjacent in G. A subset J being independent and dominating is a kernel of G.
The concept of kernels was initiated in 1953 by von Neumann and Morgenstern in digraphs with regard to game theory (see [2]). One of the pioneers studying the kernels in digraphs was C. Berge (see [3][4][5]). In literature, we can find many types and generalizations of kernels in digraphs (for results and applications, see, for example, [6][7][8][9][10][11]). The problem of the existence of kernels in undirected graphs is trivial because every maximal independent set is a kernel. Currently, distinct kind of kernels in undirected graphs are being studied quite intensively and many papers are available. For results and application, see, for example, [12][13][14][15][16][17][18]. Among many types of kernels in undirected graphs, there are kernels related to multiple domination, introduced by Fink and Jacobson in [19]. Let p ≥ 1 be an integer. A subset S is said to be p-dominating if every vertex outside S has at least p neighbors in S. If p = 1, then we obtain a dominating set in the classical sense. If p = 2, we get a 2-dominating set. A set which is 2-dominating and independent is named a 2dominating kernel ((2-d)-kernel in short). The concept of (2-d)-kernels was introduced by A. Włoch in [20]. Some properties of (2-d)-kernels were studied in [21][22][23][24]. In particular, in [23], it was proved that the problem of the existence of (2-d)-kernels is N P-complete for general graphs. In [25], Nagy extended the concept of (2-d)-kernels to k-dominating kernels. He considered a k-dominating set instead of the 2-dominating set, which he called k-dominating independent sets. Some properties of these sets were studied in [26,27].
The number of (2-d)-kernels in the graph G is denoted by σ(G). Let G be a graph with the (2-d)-kernel. The minimum cardinality of the (2-d)-kernel of G is called a lower (2-d)kernel number and denoted by γ (2−d) (G). The maximum cardinality of the (2-d)-kernel of G is called an upper (2-d)-kernel number and is denoted by Γ (2−d) (G).
In this paper, we consider two different generalizations of the Petersen graph. Various types of domination in the class of generalized Petersen graphs have been extensively studied in the literature (see [28][29][30][31][32]). Referring to this research, we will consider (2-d)kernels for two different generalizations of the Petersen graph. We solve the problem of the existence of (2-d)-kernels, their number, and their cardinality in these graphs. Moreover, we determine a lower and an upper kernel number in these graphs. It is worth noting that each of presented generalizations of the Petersen graph has a symmetric structure. This property is useful in finding (2-d)-kernels in these graphs.

Main Results
In this section, we consider the problem of the existence of (2-d)-kernels in two different generalizations of the Petersen graph. In particular, we give complete characterizations of these generalizations, which have the (2-d)-kernel. We determine the number of (2-d)-kernels in these graphs as well as the lower and the upper (2-d)-kernel number.
In the further part of the paper, we will use green color to mark vertices belonging to the (2-d)-kernel, and red color to indicate vertices that cannot belong to it.

Generalized Petersen Graph
Let n ≥ 3, k < n 2 be integers. The graph P(n, k) is called the generalized Petersen graph, if V(P(n, k)) = scripts are reduced modulo n. These graphs were first defined by Watkins in [33]. Figure 1 shows generalized Petersen graphs P(10, 3), P(5, 2) and examples of (2-d)-kernels in these graphs.  We start with the problem of existence of (2-d)-kernels. At the beginning, we give a sufficient condition, emerging from the property of bipartite graphs. We have the following complete characterization of bipartite generalized Petersen graphs. Proposition 1 ([34]). Let n ≥ 3, k < n 2 be integers. The graph P(n, k) is bipartite if and only if n is even and k is odd.
From this characterization we directly obtain the sufficient condition for the existence of (2-d)-kernels. Proposition 2. Let n ≥ 3, k < n 2 be integers. If n is even and k is odd, then the graph P(n, k) has at least two (2-d)-kernels which are a partition of the vertex set.
Proof. Let n, k be as in the statement of the proposition. From Proposition 1, it follows that the graph P(n, k) is a bipartite graph. Thus, there exist two independent sets of vertices V 1 , V 2 that are a partition of the set V(P(n, k)). Moreover, the graph P(n, k) is a 3-regular graph. Therefore, sets V 1 , V 2 are (2-d)-kernels of the graph P(n, k). Now, we improve the above proposition to obtain the complete characterization of the generalized Petersen graph having (2-d)-kernel. Theorem 1. Let n ≥ 3, k < n 2 be integers. The graph P(n, k) has a (2-d)-kernel if and only if (i) n is even and k is odd or (ii) n ≡ 0(mod 5) and k ≡ 2(mod 5) or (iii) n ≡ 0(mod 5) and k ≡ 3(mod 5).
Proof. If n = 3, 4, then the result is obvious. Let n ≥ 5, k < n 2 be integers. If n is even and k is odd, then by Proposition 2, (i) follows. Let n ≡ 0(mod 5), k ≡ j(mod 5), j = 2, 3. We will show that the set The independence of J follows from the definition of P(n, k). Let us assume that x ∈ V(P(n, k)) \ J. Then, either We consider two cases. 1.
x = u s . If s ≡ 1(mod 5), then {u s−1 , u s+1 } ⊆ N(u s ) and u s−1 , u s+1 ∈ J. If s ≡ 3(mod 5), then Summing up all the above cases we obtain that every vertex \ J is isomorphic to either P 1 or P 2 . We will show that in the graph P(n, k) having a (2-d)-kernel, the configurations of these paths P 1 , P 2 on the outer cycle, which are shown in the Figure 2 are forbidden. Let us consider the following cases. 1.
First, we will prove that the configuration of the paths P 1 , P 2 shown on the left side of the Figure 2 is forbidden. Suppose that u i , u i+3 , u i+6 ∈ J for some i, as in Figure 3.
We have the next two possibilities. Figure 4).
This means that u i+3+3k is not 2-dominated, a contradiction. +k +2k +3k Hence, for each n and k, it is not possible that the vertices u i , u i+3 , u i+6 belong to a (2-d)-kernel of P(n, k).

2.
Now, we will prove that the configuration of the paths P 1 , P 2 shown on the right side of the Figure 2 is forbidden. Suppose that u i , u i+2 , u i+4 , u i+7 ∈ J for some i, as in Figure 6. We consider four subcases. 2.1.

v i+1
, v i+3 ∈ J for some i. Proving analogously as in subcase 2.3., we obtain a contradiction with the assumption that J is a (2-d)-kernel. Therefore, for each n and k, it is not possible that the vertices u i , u i+2 , u i+4 , u i+7 belong to a (2-d)-kernel of P(n, k).
Hence, for the graph with the (2-d)-kernel, the configurations of P 1 , P 2 shown in the Figure 10 are the only ones that may be possible. Now, we will show that they are indeed possible. Figure 10. Possible configurations of the paths P 1 , P 2 for the graph P(n, k) with the (2-d)-kernel.

3.
Suppose that u i , u i+2 , u i+4 ∈ J for some i, as in Figure 11. We consider four subcases. 3.1.

3.3.
v i+1 ∈ J and v i+3 / ∈ J for some i. Proving analogously as in Section 3.2., we obtain a contradiction with the assumption that J is a (2-d)-kernel. 3.4. v i+1 , v i+3 ∈ J for some i (see Figure 14).
+k +2k This means that v i+k , v i−k ∈ J and also v i+2+k , v i+4+k , u i+1+k , u i+3+k belong to a (2-d)kernel (see Figure 15). +k Figure 15. The case when Hence, n must be even, and from the definition of P(n, k), we conclude that k must be odd, which proves (i).
First, we prove that v i+1 / ∈ J. Suppose on contrary that v i+1 ∈ J, as in Figure 16.
Hence, n must be divisible by 5, and from the definition of P(n, k), we conclude that k ≡ 2(mod 5), which proves (ii).
Hence, n must be divisible by 5, and from the definition of P(n, k), we conclude that k ≡ 3(mod 5), which proves (iii), which ends the proof.
Basing on the proof of Theorem 1, the following corollaries are obtained. They concern the number of (2-d)-kernels in the generalized Petersen graph as well as the lower and upper (2-d)-kernel numbers. By a rotation of configurations shown on Figure 10, condition (i) of Theorem 1 gives two (2-d)-kernels in generalized Petersen graph and conditions (ii) and (iii) give five (2-d)-kernels. Therefore, if n and k satisfy more than one of these conditions, we obtain more (2-d)-kernels. Moreover, the proof of the Theorem 1 presents the constructions of the (2-d)-kernels in the generalized Petersen graph P(n, k). Corollary 1. Let n ≥ 3, k < n 2 be integers. Then, for n ≡ 0(mod 10) and k ≡ a(mod 10), a = 3, 7 5 for n ≡ 5(mod 10) and k ≡ a(mod 5), a = 2, 3 or for n ≡ 0(mod 10) and k ≡ a(mod 10), a = 2, 8 2 for n ≡ 0(mod 10) and k ≡ a(mod 10), a = 1, 5, 9 or for even n, n ≡ 0(mod 10) and odd k.
The above corollaries characterize all possible graphs P(n, k), which have the (2-d)kernel.

The Second Generalization of the Petersen Graph
Now, we consider another generalization of the Petersen graph. Let n ≥ 5 be an integer. Let C n be a cycle and C n its complement such that V(C n ) = {x 1 , x 2 , . . . , x n }, V(C n ) = {x c 1 , x c 2 , . . . , x c n } with the numbering of vertices in the natural order. Let G(n) be the graph such that V(G(n)) = V(C n ) ∪ V(C n ) and E(G(n)) = E(C n ) ∪ E(C n ) ∪ {x i x c i ; i ∈ {1, 2, . . . , n}}. Figure 21 shows an example of a (2-d)-kernel in G (13). It is easy to check that if n = 5, then G(5) is isomorphic to the Petersen graph. Proof. Let n ≥ 5 be odd. We will show that J = {x c 2 , x c 3 , x 1 , x 4 , x 6 , . . . , x n−1 } is the (2-d)kernel of the graph G(n). The independence of J is obvious. It is sufficient to show that J is a 2-dominating set. By the definition of the graph G(n), we can assume that x n+1 = x 1 . Suppose that y ∈ V(G(n)) \ J. Hence, y ∈ V(C n ) or y ∈ V(C n ). Let y ∈ V(C n ). Thus y = x k , k ∈ {2, 3, 5, . . . , n}. If x c k / ∈ J, then there exist vertices x k−1 , x k+1 ∈ J adjacent to x k . If x c k ∈ J, then k = 2 or k = 3. For k = 2, the vertex x 2 is adjacent to x 1 , x c 2 ∈ J. Moreover, if k = 3, then the vertex x 3 is adjacent to x 4 , x c 3 ∈ J. Hence, every vertex from the set V(C n ) is 2-dominated by the set J. Let now y ∈ V(C n ). Thus y = x c k , k ∈ {1, 4, 5, . . . , n}. Then, the vertex x c k , k ∈ {5, 6, . . . , n} is adjacent to x c 2 , x c 3 ∈ J. If k = 1, then x c 1 x 1 , x c 1 x c 3 ∈ E(G(n)). Moreover, for k = 4 the vertex x c 4 is adjacent to x 4 , x c 2 . Therefore, vertices from the set V(C n ) are 2-dominated by J and hence J is a (2-d)-kernel of G(n).

G(13)
Conversely, suppose that a graph G(n) has a (2-d)-kernel J. We will show that n is odd. By the definition of the graph G(n), we obtain that J ∩ V(C n ) = ∅. Otherwise, vertices from the set V(C n ) are not 2-dominated by the set J. Let x c 1 ∈ J. Then either x c 2 ∈ J or x c n ∈ J. Otherwise, x c 2 or x c n is not 2-dominated. Hence, |J ∩ V(C n )| = 2. Without loss of generality assume that x c 1 , x c 2 ∈ J. This means that x c i , i ∈ {4, 5, . . . , n − 1} is 2-dominated by J and x c 3 , x c n are dominated by J.
Finally, it turns out that if a graph G(n) has (2-d)-kernel, then the number of (2-d)kernels depends linearly on the number of vertices. Moreover, each (2-d)-kernel of G(n) has the same cardinality.
Proof. Let n ≥ 5 be odd. From the construction of a (2-d)-kernel described in the proof of Theorem 2, we conclude that exactly two not adjacent vertices from the set V(C n ) ⊂ V(G(n)) belong to a (2-d)-kernel. The selection of these two vertices will determine the (2-d)-kernel in G(n). Since two not adjacent vertices can be chosen on n ways, σ(G(n)) = n. Moreover, from the construction of (2-d)-kernels in G(n), it follows that all of them have the same cardinality. Hence, γ (2−d) (G(n)) = Γ (2−d) (G(n)) = n 2 + 2, which ends the proof.

Concluding Remarks
In this paper, we considered two different generalizations of the Petersen graph, and we discussed the problem of the existence of (2-d)-kernels in these graphs. In particular, we determined the number of (2-d)-kernels in these graphs and their lower and upper (2-d)-kernel number. The generalized Petersen graphs considered in this paper are special cases of I-graphs (see, for example, [35]). The I-graph I(n, j, k) is a graph with a vertex set V(I(n, j, k)) = {u 1 , u 2 , . . . , u n , v 1 , v 2 , . . . , v n } and an edge set E(I(n, j, k)) = {u i u i+j , u i v i , v i v i+k ; i ∈ {1, 2, . . . , n}}, where subscripts are reduced modulo n. Because P(n, k) = I(n, 1, k), the results obtained could be a starting point to studying and counting (2-d)-kernels in I-graphs. It could also be interesting to investigate the number of (2-d)-kernels in other generalizations of generalized Petersen graphs. For more generalizations, see, for example, [36].
Author Contributions: Both authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.