The g-extra connectivity of the Mycielskian

TL;DR

This paper explores the g-extra connectivity of the Mycielskian graph, establishing a precise relationship between the connectivity of the original graph and its Mycielskian, and providing bounds for g-extra connectivity.

Contribution

It derives a formula linking the g-extra connectivity of a graph and its Mycielskian, advancing understanding of graph robustness and connectivity properties.

Findings

Proves that rac{2g+1}{ ext{Mycielskian}} ext{ connectivity} = 2 imes ext{original connectivity} + 1.

Establishes bounds for g-extra connectivity as rac{g+1}{ ext{original graph}} ext{ and } ext{floor}(n/2).

Provides insights into the structural properties of Mycielskian graphs related to connectivity.

Abstract

The $g$-extra connectivity is an important parameter to measure the ability of tolerance and reliability of interconnection networks. Given a connected graph $G=(V,E)$ and a non-negative integer $g$, a subset $S\subseteq V$ is called a $g$-extra cut of $G$ if $G-S$ is disconnected and every component of $G-S$ has at least $g+1$ vertices. The cardinality of the minimum $g$-extra cut is defined as the $g$-extra connectivity of $G$, denoted by $\kappa_g(G)$. In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph $G$ into a new graph $\mu(G)$, which is called the Mycielskian of $G$. This paper investigates the relationship of the g-extra connectivity of the Mycielskian $\mu(G)$ and the graph $G$, moreover, show that $\kappa_{2g+1}(\mu(G))=2\kappa_{g}(G)+1$ for $g\geq 1$ and $\kappa_{g}(G)\leq min\{g+1, \lfloor\frac{n}{2}\rfloor\}$.