On corona of Konig-Egervary graphs

TL;DR

This paper characterizes graphs whose coronas are 0- or 1-König-Egerváry graphs, expanding understanding of the structure of such graphs in relation to their maximum independent sets and matchings.

Contribution

It provides a complete characterization of graphs with coronas that are 0- or 1-König-Egerváry graphs, a novel structural insight in graph theory.

Findings

Characterization of graphs with corona as 0-König-Egerváry graphs

Characterization of graphs with corona as 1-König-Egerváry graphs

Extension of known properties of König-Egerváry graphs

Abstract

Let $\alpha(G)$ denote the cardinality of a maximum independent set and $\mu(G)$ be the size of a maximum matching of a graph $G=\left( V,E\right) $. If $\alpha(G)+\mu(G)=\left\vert V\right\vert $, then $G$ is a K\"{o}nig-Egerv\'{a}ry graph, and $G$ is a $1$-K\"{o}nig-Egerv\'{a}ry graph whenever $\alpha(G)+\mu(G)=\left\vert V\right\vert -1$. The corona $H\circ\mathcal{X}$ of a graph $H$ and a family of graphs $\mathcal{X}=\left\{ X_{i}:1\leq i\leq\left\vert V(H)\right\vert \right\} $ is obtained by joining each vertex $v_{i}$ of $H$ to all the vertices of the corresponding graph $X_{i},i=1,2,...,\left\vert V(H)\right\vert $. In this paper we completely characterize graphs whose coronas are $k$-K\"{o}nig-Egerv\'{a}ry graphs, where $k\in\left\{ 0,1\right\} $.