The Roller-Coaster Conjecture Revisited

A graph is well-covered if all its maximal independent sets are of the same cardinality [25]. If G is a well-covered graph, has at least two vertices, and G − v is well-covered for every vertex v, then G is a 1-well-covered graph [26]. We call G a λ-quasi-regularizable graph if λ · |S| ≤ |N (S)| for every independent set S of G. The independence polynomial I(G;x) is the generating function of independent sets in a graph G [9]. The Roller-Coaster Conjecture [24], saying that for every permutation σ of the set { ⌈