Relating the independence number and the dissociation number

Abstract

AbstractThe independence number and the dissociation number of a graph are the largest orders of induced subgraphs of of maximum degree at most 0 and at most 1, respectively. We consider possible improvements of the obvious inequality . For connected cubic graphs distinct from , we show , and describe the rich and interesting structure of the extremal graphs in detail. For bipartite graphs, and, more generally, triangle‐free graphs, we also obtain improvements. For subcubic graphs though, the inequality cannot be improved in general, and we characterize all extremal subcubic graphs.