A note on independent sets in sparse-dense graphs
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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 𝒮 and 𝒟, where 𝒟 is a subclass of the complement of K_t-free graphs (for some t), and graphs in 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 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 ℓ cliques, so-called (k, ℓ)-graphs. concerning the values of k and ℓ of (k, ℓ)-graphs. A P-time algorithm that does not require (1,ℓ)-partitions for determining whether a (1,ℓ)-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.
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