Common Independence in Graphs

Magda Dettlaff,Jerzy Topp,Magdalena Lemańska

doi:10.3390/sym13081411

Abstract

The cardinality of a largest independent set of G, denoted by α(G), is called the independence number of G. The independent domination number i(G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by αc(G), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with |X|≥r. The common independence number αc(G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc(G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i(G)≤αc(G)≤α(G). In this paper, we characterize the trees T for which i(T)=αc(T), and the block graphs G for which αc(G)=α(G).

Highlights

We introduce the concept of the common independence number of a graph G, denoted by αc ( G ), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with | X | ≥ r
G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc ( G ) in G, and there are vertices in G that do not belong to any larger independent set in G
For our studies of the common independence number of a graph, we begin from straightforward propositions and simple examples

Summary

Introduction

The cardinality of a largest (i.e., maximum) independent set of G, denoted by α( G ), is called the independence number of G. We introduce the concept of the common independence number of a graph G, denoted by αc ( G ), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with | X | ≥ r. G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc ( G ) in G, and there are vertices in G that do not belong to any larger independent set in G.

Preliminaries

Open Problems