Vertex percolation on expander graphs

Abstract

We say that a graph G = ( V , E ) on n vertices is a β -expander for some constant β > 0 if every U ⊆ V of cardinality | U | ≤ n 2 satisfies | N G ( U ) | ≥ β | U | where N G ( U ) denotes the neighborhood of U . In this work we explore the process of deleting vertices of a β -expander independently at random with probability n − α for some constant α > 0 , and study the properties of the resulting graph. Our main result states that as n tends to infinity, the deletion process performed on a β -expander graph of bounded degree will result with high probability in a graph composed of a giant component containing n − o ( n ) vertices that is in itself an expander graph, and constant size components. We proceed by applying the main result to expander graphs with a positive spectral gap. In the particular case of ( n , d , λ ) -graphs, that are such expanders, we compute the values of α , under additional constraints on the graph, for which with high probability the resulting graph will stay connected, or will be composed of a giant component and isolated vertices. As a graph sampled from the uniform probability space of d -regular graphs with high probability is an expander and meets the additional constraints, this result strengthens a recent result due to Greenhill, Holt and Wormald about vertex percolation on random d -regular graphs. We conclude by showing that performing the above described deletion process on graphs that expand sub-linear sets by an unbounded expansion ratio, with high probability results in a connected expander graph.