On the Core of a Unicyclic Graph

TL;DR

This paper investigates the structure of the core in unicyclic graphs, establishing conditions under which the core equals the union of the cores of the trees obtained by removing the cycle.

Contribution

It proves that for certain unicyclic graphs, the core can be characterized as the union of the cores of the component trees after removing the cycle.

Findings

Core of unicyclic graph equals union of tree cores under specific conditions

Provides a characterization of the core in unicyclic graphs

Connects independence number and maximum matching in the analysis

Abstract

A set S is independent in a graph G if no two vertices from S are adjacent. By core(G) we mean the intersection of all maximum independent sets. The independence number alpha(G) is the cardinality of a maximum independent set, while mu(G) is the size of a maximum matching in G. A connected graph having only one cycle, say C, is a unicyclic graph. In this paper we prove that if G is a unicyclic graph of order n and n-1 = alpha(G) + mu(G), then core(G) coincides with the union of cores of all trees in G-C.