Maximal and maximum dissociation sets in general and triangle-free graphs

•Decide the maximum number of the maximal and maximum dissociation set in general graphs.•Decide the maximum number of the maximal and maximum dissociation set in triangle-free graphs.•Characterize the extremal graphs on which these upper bounds are attained. In a graph G, a subset of vertices is a dissociation set if it induces a subgraph with maximum degree at most 1. A maximal dissociation set of G is a dissociation set which is not a proper subset of any other dissociation sets. A maximum dissociation set is a dissociation set of maximum size. We show that every graph of order n has at most 10n5 maximal dissociation sets, and that every triangle-free graph of order n has at most 6n4 maximal dissociation sets. We also characterize the extremal graphs on which these upper bounds are attained.