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

Abstract

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.

On Proper 2-Dominating Sets in Graphs

“…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.…”