A simple proof that finding a maximal independent set in a graph is in $\mathcal{NC}$

Abstract

An independent set in a graph G is a set $I\subseteq V(G)$ such that the induced subgraph on I has no edges. A maximal independent set (MIS) is an independent set not properly contained in any other independent set. We present a simple proof that an algorithm of Luby [SIAM J. Comput. 15 (1986) 1036] is an $\mathcal{RNC}$ algorithm for finding an MIS in a graph. Our proof can be easily derandomized, giving a corresponding $\mathcal{NC}$ algorithm.