Super Connectivity of the Generalized Mycielskian of Graphs

Introduction

All graphs considered in this paper are simple, finite, nontrivial and undirected.

Let G be a graph with vertex set $V^0=\{v_0^0,v_1^0,\ldots,v_{n-1}^0\}$ and edge set E ⁰. Given an integer m ≥ 1, the m-Mycielskian (also known as the generalized Mycielskian) of G, denoted by μ _m (G), is the graph whose vertex set is the disjoint union

$$V^0\cup V^1\cup\ldots\cup V^m\cup\{u\},$$

where $V^i=\{v_j^i;v_j^0\in V^0\}$ is the i-th copy of V ⁰, i = 1,2,…,m, and edge set

$$E^0\cup\Big(\mathop{\cup}\limits_{i=0}^{m-1}\{v_j^iv_{j'} ^{i+1}\ :v_j^0v_{j'}^0\in E^0\}\Big)\cup \{v_j^mu: v_j^m\in V^m\}.$$