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

Paweł Bednarz,Natalia Paja

doi:10.3390/sym13101948

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.

Highlights

We use the standard terminology and notation of graph theory
If G ⊆ G and G contain all the edges xy ∈ E with x, y ∈ V, G is an induced subgraph of G and we write G := V G
We consider the problem of the existence of (2-d)-kernels in two different generalizations of the Petersen graph

Summary

Introduction

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). A subset S ⊆ V(G) is called an independent set of G if no two vertices of S are adjacent in G. The problem of the existence of kernels in undirected graphs is trivial because every maximal independent set is a kernel. 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. Various types of domination in the class of generalized Petersen graphs have been extensively studied in the literature (see [28,29,30,31,32]). 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

Generalized Petersen Graph

The Second Generalization of the Petersen Graph

Concluding Remarks