let $g$ be a simple graph. an independent set is a set ofpairwise non-adjacent vertices. the number of vertices in a maximum independent set of $g$ isdenoted by $alpha(g)$. in this paper, we characterize graphs $g$ with $n$ vertices and with maximumnumber of maximum independent sets provided that $alpha(g)leq 2$ or $alpha(g)geq n-3$.

We study the Maximum Independent Set of Rectangles (MISR) problem: given a collection R of n axis-parallel rectangles, find a maximum-cardinality subset of disjoint rectangles. MISR is a special case of the classical Maximum Independent Set problem, where the input is restricted to intersection graphs of axis-parallel rectangles. Due to its many applications, ranging from map labeling to data m...

We study the algorithmic task of finding a large independent set in sparse Erdős– Rényi random graph with $n$ vertices and average degree $d$. The maximum is known to have size $(2 \log d / d)n$ double limit $n \to \infty$ followed by $d \infty$, but best polynomial-time algorithms can only find an half-optimal $(\log d)n$. show that class low-degree polynomial sets no larger, improving upon re...