Robust Domination in Random Graphs

TL;DR

This paper investigates the properties of robust dominating sets in random graphs, focusing on their resilience to edge removals and establishing conditions for their optimality using probabilistic methods.

Contribution

It introduces a new concept of robustness in dominating sets for random graphs and provides asymptotic conditions for their optimality, especially in sparse graph regimes.

Findings

Derived sufficient conditions for asymptotic optimality of robust domination number

Analyzed robust domination in sparse random graphs with linear edge growth

Applied probabilistic and martingale techniques to study robustness properties

Abstract

In this paper, we study "robust" dominating sets of random graphs that retain the domination property even if a small \emph{deterministic} set of edges are removed. We motivate our study by illustrating with examples from wireless networks in harsh environments. We then use the probabilistic method and martingale difference techniques to determine sufficient conditions for the asymptotic optimality of the robust domination number. We also discuss robust domination in sparse random graphs where the number of edges grows at most linearly in the number of vertices.