S S symmetry Article On (2-d)-Kernels in the Tensor Product of Graphs Paweł Bednarz The Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, Aleja Powsta´nców Warszawy 12, 35-959 Rzeszów, Poland; [email protected] Abstract: In this paper, we study the existence, construction and number of (2-d)-kernels in the tensor product of paths, cycles and complete graphs. The symmetric distribution of (2-d)-kernels in these products helps us to characterize them. Among others, we show that the existence of (2-d)-kernels in the tensor product does not require the existence of a (2-d)-kernel in their factors. Moreover, we determine the number of (2-d)-kernels in the tensor product of certain factors using Padovan and Perrin numbers. Keywords: domination; independence; (2-d)-kernel number; tensor product MSC: 05C69; 05C76 1. Introduction and Preliminary Results In general, we use the standard terminology and notation of graph theory; see [1]. Only undirected, simple graphs are considered. By Pn, Cn and Kn we mean a path, cycle and complete graph on n vertices, respectively. Moreover, Kn,m is a complete bipartite + graph on n m vertices. A vertex of degree one is called a leaf, and the set of all leaves of a graph G is denoted by L(G). Citation: Bednarz, P. On Let G and H be two disjoint graphs. The tensor product of two graphs G and H is the (2-d)-Kernels in the Tensor graph G × H such that V(G × H) = V(G) × V(H) and E(G × H) = f(xi, yp)(xj, yq); xixj 2 Product of Graphs. Symmetry 2021, 13, E(G) and ypyq 2 E(H)g. The tensor product is also called a direct product or a categorical 230. https://doi.org/10.3390/ product; see [2] for the details. sym13020230 We say that D ⊆ V(G) is a dominating set of G, if every vertex of G is either in D or is adjacent to at least one vertex of D. A subset S of vertices is an independent set of G if no Academic Editors: Alice Miller and two vertices of S are adjacent in G. Charles F. Dunkl A subset J is a kernel of G if J is independent and dominating. The topic of kernels Received: 8 December 2020 was introduced by von Neumann and Morgenstern in digraphs in the 1930s in the context Accepted: 27 January 2021 of game theory; see [3]. Published: 30 January 2021 Since then, kernels have been intensively studied in graph theory for their relations Publisher’s Note: MDPI stays neu- to various problems, such as list colourings and perfectness. C. Berge made a great tral with regard to jurisdictional clai- contribution to the theory of kernels in digraphs, being one of the pioneers studying the ms in published maps and institutio- problem of the existence of kernels in digraphs and using a kernel for solving problems nal affiliations. in other areas of mathematics; see [4–6]. In the literature, we can find many types and generalizations of kernels in digraphs, where the existence problems are studied, and some interesting results can be found in [7–11]. In an undirected graph, every maximal independent set is a kernel, so the existence Copyright: © 2021 by the author. Li- problem in an undirected graph is trivial. The problem becomes more complicated when censee MDPI, Basel, Switzerland. restrictions related to the domination or the independence are added. In that way, many This article is an open access article interesting types of kernels in undirected graphs were introduced and studied (for example, distributed under the terms and con- (k,l)-kernels [12–14], efficient dominating sets [15], secondary independent dominating ditions of the Creative Commons At- sets [16,17], restrained independent dominating sets [18], strong (1,1,2)-kernels [19] and tribution (CC BY) license (https:// others). Among many types of kernels in undirected graphs, there are kernels related to creativecommons.org/licenses/by/ 4.0/). multiple domination. The concept of multiple domination was introduced by J. F. Fink Symmetry 2021, 13, 230. https://doi.org/10.3390/sym13020230 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 230 2 of 9 and M. S. Jacobson in [20]. Recall that for an integer p ≥ 1, a subset D ⊆ V(G) is called a p-dominating set of G if every vertex from V(G) n D has at least p neighbours in D. If p = 1, then we get the classical definition of the dominating set. If p = 2, then we get the two-dominating set. Based on this definition in [21], A. Włoch introduced and studied the concept of a two-dominating kernel ((2-d)-kernel). A subset J ⊆ V(G) is a (2-d)-kernel of G if J is independent and two-dominating. The number of (2-d)-kernels in a graph G is denoted by s(G). Let G be a graph with a (2-d)-kernel. The minimum (resp. maximum) cardinality of a (2-d)-kernel of G is called a lower (resp. upper) (2-d)-kernel number, and it is denoted by j(G) (resp. J(G)). Existence problems of (2-d)-kernels were next considered in [22–24]. Among others, in [24], it was proven that the problem of the existence of (2-d)-kernels is NP-complete for general graphs. This paper is a continuation of those considerations. Referring to the definition of the (2-d)-kernel and papers [21–24], Z. L. Nagy extended the concept of (2-d)-kernels to k-dominating kernels (named as k-dominating independent sets) by considering a k-dominating set instead of the two-dominating set, and that type of kernels was studied in [25–27]. However, it is worth noting that the existence of k-dominating kernels requires a sufficiently large degree of vertices, significantly reducing the number of graphs with k-dominating kernels. Consequently, it is interesting to limit the requirements for the degrees of vertices if existing problems are considered. Graph products are useful tools for obtaining new classes of graphs that can be described based on factor properties. In kernel theory, the existence problems are very often studied in products of graphs; see for example [17,19]. For (2-d)-kernels in the Cartesian product, some results were obtained in [23]. This paper considers the problem of the existence of (2-d)-kernels and their number in the tensor product of graphs. The problem of counting various sets (dominating, independent, kernels) in graphs and their relations to the numbers of the Fibonacci type were studied in [11,24,28]. Our results are as follows. • We partially solve the problem of the existence of (2-d)-kernels in the tensor product, in particular giving the complete solutions for cases where factors are paths, cycles or complete graphs. • We obtain new interpretations of the Padovan and Perrin numbers related to (2-d)- kernels in the tensor product of basic classes of graphs, i.e., paths, cycles and complete graphs. 2. (2-d)-Kernels in the Tensor Product of Graphs In this section, we consider the problem of the existence and the number of (2-d)- kernels in the tensor product of two graphs. We will use the following results of the tensor product of two graphs, which has been obtained in [2,29]. Theorem 1 ([29]). Let G and H be connected non-trivial graphs. If at least one of them has an odd cycle, then G × H is connected. If both G and H are bipartite, then G × H has exactly two connected components. Theorem 2 ([2]). Let G and H be graphs. If G is bipartite, then G × H is bipartite. The following theorems, concerning the problem of the existence of (2-d)-kernels in paths and bipartite graphs, were proven in [21]. We use them in the rest of the paper. Theorem 3 ([21]). The graph Pn has a (2-d)-kernel J if and only if n = 2p + 1, p ≥ 1, and this kernel is unique. If it holds, then jJj = p + 1. Symmetry 2021, 13, 230 3 of 9 Theorem 4 ([21]). Let G = G(V1, V2) be a bipartite graph. If for each two vertices x, y 2 L(G), dG(x, y) ≡ 0(mod 2), then G has a (2-d)-kernel. The simple observation of bipartite graphs leads us to the following result. Corollary 1. Let G = G(V1, V2) be a bipartite graph. If d(G) ≥ 2, then G has a (2-d)-kernel. We present results concerning the existence problems of (2-d)-kernels, as well as their number in the tensor product of paths, cycles and complete graphs. Note that tensor products of these graphs have a symmetric structure. This property is very useful in finding (2-d)-kernels in these products. Moreover, we can notice that the kernels themselves are distributed symmetrically in tensor products. It turns out that even in the simplest cases of the tensor product of graphs, the problem of the existence of (2-d)-kernels is not an easy one. Theorem3 tells us that there are not any (2-d)-kernels in the path with an even number of vertices. When studying the tensor product of two paths, a similar result was found, namely the (2-d)-kernel exists if and only if both paths do not have even number of vertices simultaneously. We prove this in the following Theorem. Theorem 5. Let n, m ≥ 1 be integers. The graph Pn × Pm has a (2-d)-kernel if and only if at least one of the numbers n or m is odd.