The g-extra connectivity of the Mycielskian

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-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 κ 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 µ(G), which is called the Mycielskian of G. This paper investigates the relationship of the g-extra connectivity of the Mycielskian µ(G) and the graph G, moreover, show that κ 2g+1 (µ(G)) = 2κ g (G) + 1 for g ≥ 1 and κ g (G) ≤ min{g + 1, ⌊ n 2 ⌋}.