On the local resilience of random geometric graphs with respect to connectivity and long cycles

TL;DR

This paper investigates the local resilience of random geometric graphs in maintaining connectivity, long cycles, and Hamiltonicity when a fraction of neighbors are removed, revealing thresholds and limitations in these properties.

Contribution

It introduces the concept of local resilience in random geometric graphs for connectivity and cycle properties, providing thresholds and resilience bounds that differ from binomial random graphs.

Findings

Random geometric graphs are $(1/2- ext{small } ext{epsilon})$-resilient for $k$-connectivity above the connectivity threshold.

Connectivity is not $(1/2- ext{small } ext{epsilon})$-resilient in 2D for small epsilon.

Graphs are $(1/2- ext{epsilon})$-resilient for containing long cycles above the threshold.

Abstract

Given an increasing graph property $\mathcal{P}$, a graph $G$ is $\alpha$-resilient with respect to $\mathcal{P}$ if, for every spanning subgraph $H\subseteq G$ where each vertex keeps more than a $(1-\alpha)$-proportion of its neighbours, $H$ has property $\mathcal{P}$. We study the above notion of local resilience with $G$ being a random geometric graph $G_d(n,r)$ obtained by embedding $n$ vertices independently and uniformly at random in $[0,1]^d$, and connecting two vertices by an edge if the distance between them is at most $r$. First, we focus on connectivity. We show that, for every $\varepsilon>0$, for $r$ a constant factor above the sharp threshold for connectivity $r_c$ of $G_d(n,r)$, the random geometric graph is $(1/2-\varepsilon)$-resilient for the property of being $k$-connected, with $k$ of the same order as the expected degree. However, contrary to binomial random graphs, for sufficiently small $\varepsilon>0$, connectivity is not born $(1/2-\varepsilon)$-resilient in $2$-dimensional random geometric graphs. Second, we study local resilience with respect to the property of containing long cycles. We show that, for $r$ a constant factor above $r_c$, $G_d(n,r)$ is $(1/2-\varepsilon)$-resilient with respect to containing cycles of all lengths between constant and $2n/3$. Proving $(1/2-\varepsilon)$-resilience for Hamiltonicity remains elusive with our techniques. Nevertheless, we show that $G_d(n,r)$ is $\alpha$-resilient with respect to Hamiltonicity for a fixed constant $\alpha = \alpha(d)<1/2$.