Neighbor connectivity of pancake graphs and burnt pancake graphs

Abstract

The neighbor connectivity (resp. edge neighbor connectivity) of a graph G is the least number of vertices (resp. edges) such that removing their closed neighborhoods from G results in a graph that is empty, complete (resp. trivial), or disconnected. For a graph G , the neighbor connectivity and the edge neighbor connectivity of G are denoted by κ N B ( G ) and λ N B ( G ) , respectively. The notion of these two kinds of connectivity was derived from assessing the impact of subversion caused by the underground resistance movement in spy networks. Currently, it can provide more accurate measures regarding the reliability and fault-tolerance of networks. In this paper, we completely determine the neighbor connectivity and edge neighbor connectivity of pancake graphs and burnt pancake graphs as follows: κ N B ( P n ) = λ N B ( P n ) = n − 1 for n ≥ 3 , and κ N B ( BP n ) = λ N B ( BP n ) = n for n ≥ 2 .