Maximal and Maximum Independent Sets In Graphs With At Most r Cycles

TL;DR

This paper fully determines the maximum number of maximal independent sets in connected graphs with a limited number of cycles, completing previous partial results and characterizing extremal graphs.

Contribution

It provides a complete characterization of c(n,r) for all n and r, resolving open problems and describing extremal graphs for maximum independent sets.

Findings

Complete formula for c(n,r) for all n and r

Characterization of extremal graphs achieving c(n,r)

Resolution of open problems in maximum independent sets

Abstract

Let m(G) denote the number of maximal independent sets of vertices in a graph G and let c(n,r) be the maximum value of m(G) over all connected graphs with n vertices and at most r cycles. A theorem of Griggs, Grinstead, and Guichard gives a formula for c(n,r) when r is large relative to n, while a theorem of Goh, Koh, Sagan, and Vatter does the same when r is small relative to n. We complete the determination of c(n,r) for all n and r and characterize the extremal graphs. Problems for maximum independent sets are also completely resolved.