A note on independent sets in sparse-dense graphs

TL;DR

This paper explores the complexity of finding independent sets in graphs with sparse-dense partitions, providing polynomial-time algorithms and complexity dichotomies for specific graph classes and problems.

Contribution

It establishes conditions under which enumerating maximum independent sets is polynomial-time feasible in sparse-dense graphs, and characterizes complexity boundaries for related problems.

Findings

Enumeration of maximum independent sets is polynomial-time in certain sparse-dense graphs.

A P vs NP-hard dichotomy for maximum independent set in (k, l)-graphs.

Polynomial-time algorithms for well-coveredness in specific graph classes.

Abstract

Sparse-dense partitions was introduced by Feder, Hell, Klein, and Motwani [STOC 1999, SIDMA 2003] as a tool to solve partitioning problems. In this paper, the following result concerning independent sets in graphs having sparse-dense partitions is presented: if a $n$-vertex graph $G$ admits a sparse-dense partition concerning classes $\mathcal S$ and $\mathcal D$, where $\mathcal D$ is a subclass of the complement of $K_t$-free graphs (for some ~$t$), and graphs in $\mathcal S$ can be recognized in polynomial time, then: enumerate all maximal independent sets of $G$ (or find its maximum) can be performed in $n^{O(1)}$ time whenever it can be done in polynomial time for graphs in the class $\mathcal S$. This result has the following interesting implications: A P versus NP-hard dichotomy for Max. Independent Set on graphs whose vertex set can be partitioned into $k$ independent sets and $\ell$ cliques, so-called $(k, \ell)$-graphs. concerning the values of $k$ and $\ell$ of $(k, \ell)$-graphs. A P-time algorithm that does not require $(1,\ell)$-partitions for determining whether a $(1,\ell)$-graph $G$ is well-covered. Well-covered graphs are graphs in which every maximal independent set has the same cardinality. The characterization of conflict graph classes for which the conflict version of a P-time graph problem is still in P assuming such classes. Conflict versions of graph problems ask for solutions avoiding pairs of conflicting elements (vertices or edges) described in conflict graphs.