Abstract
In 2008, Hedetniemi et al. introduced (1,k)-domination in graphs. The research on this concept was extended to the problem of existence of independent (1,k)-dominating sets, which is an NP-complete problem. In this paper, we consider independent (1,1)- and (1,2)-dominating sets, which we name as (1,1)-kernels and (1,2)-kernels, respectively. We obtain a complete characterization of generalized corona of graphs and G-join of graphs, which have such kernels. Moreover, we determine some graph parameters related to these sets, such as the number and the cardinality. In general, graph products considered in this paper have an asymmetric structure, contrary to other many well-known graph products (Cartesian, tensor, strong).
Highlights
Introduction and Preliminary ResultsIn general, we will use the standard terminology and notation of graph theory
If we place some additional restrictions on the subset of vertices, modifying the classical concepts of domination or independence, the problem of the existence becomes more complicated
We provide the definition of the G-join of graphs
Summary
We will use the standard terminology and notation of graph theory (see [1]). The issue of the existence of kernels in undirected graphs is trivial, since every maximal independent set is a kernel. For a positive integer k, we say that the set D ∈ V(G) is (1, k)-dominating if for every x ∈ V(G) \ D there exist distinct vertices v, w ∈ D such that xv ∈ E(G) and dG(x, w) ≤ k By combining this type of domination with independence, we obtain (1, k)-kernels, i.e., subsets, which are independent and (1, k)-dominating. Problems of independence and domination in the generalized corona of graphs were considered in [10,27,28]. We give necessary and sufficient conditions for the existence of (1, 1)-kernels in the generalized corona of graphs, where all of the graphs from the sequence hn are nontrivial.
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