Strong (1, 1, 2)-Kernels in the Corona of Graphs and Some Realization Problems

Abstract

In this paper, we give the necessary and sufficient conditions for the existence of strong (1, 1, 2)-kernels in the corona of graphs. Moreover, we consider lower and upper strong (1, 1, 2)-kernel numbers and we prove that the difference between these parameters can be arbitrarily large.

On (2-d)-Kernels in the Tensor Product of Graphs

“…The problem becomes more complicated when restrictions related to the domination or the independence are added. In that way, many interesting types of kernels in undirected graphs were introduced and studied (for example, (k,l)-kernels [12][13][14], efficient dominating sets [15], secondary independent dominating sets [16,17], restrained independent dominating sets [18], strong (1,1,2)-kernels [19] and others). Among many types of kernels in undirected graphs, there are kernels related to multiple domination.…”

“…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].…”

Fibonacci numbers in graphs with strong (1, 1, 2)-kernels

“…Consequently, it seems to be interesting to limit requirements for degrees of vertices by considering secondary domination. This point of view leads to the concept of (1, 1, 2)-kernels which was introduced in [5] and was studied in [4].…”

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

“…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].…”