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On corona of Konig-Egervary graphs

Vadim E. Levit, Eugen Mandrescu

Abstract

Let $α(G)$ denote the cardinality of a maximum independent set and $μ(G)$ be the size of a maximum matching of a graph $G=\left( V,E\right) $. If $α(G)+μ(G)=\left\vert V\right\vert $, then $G$ is a König-Egerváry graph, and $G$ is a $1$-König-Egerváry graph whenever $α(G)+μ(G)=\left\vert V\right\vert -1$. 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 $k$-König-Egerváry graphs, where $k\in\left\{ 0,1\right\} $.

On corona of Konig-Egervary graphs

Abstract

Let denote the cardinality of a maximum independent set and be the size of a maximum matching of a graph . If , then is a König-Egerváry graph, and is a -König-Egerváry graph whenever . The corona of a graph and a family of graphs is obtained by joining each vertex of to all the vertices of the corresponding graph . In this paper we completely characterize graphs whose coronas are -König-Egerváry graphs, where .

Paper Structure

This paper contains 3 sections, 12 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Conclusions

Key Result

Theorem 2.1

Let $G=H\circ\left\{ X_{i}:1\leq i\leq n\left( H\right) \right\}$ and $F=\left\{ v_{i}\in V\left( H\right) :\mu(X_{i})=\mu (v_{i}\circ X_{i})\right\}$. (i) $\alpha(G)={\sum\limits_{i=1}^{n\left( H\right) }} \alpha(X_{i})$; (ii) $\mu(G)=\mu\left( H\left[ F\right] \right) +{\sum\limits_{i=1

Figures (3)

  • Figure 1: $G_{1}-v_{1}$, $G_{2}-e_{2}$ are König-Egerváry graphs, while $G_{1}-v_{2}$ and $G_{2}-e_{1}$ are not König-Egerváry graphs.
  • Figure 2: $G=(K_{3}+v_{1}v_{3})\circ\{K_{3},K_{2},K_{3},2K_{1}\}$ is not a König-Egerváry graph.
  • Figure 3: $G_{1}$ is an $1$-König-Egerváry graph, while $G_{2}$ is a König-Egerváry graph.

Theorems & Definitions (13)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Corollary 2.7
  • Corollary 2.8
  • Lemma 2.9
  • Theorem 2.10
  • ...and 3 more