Relating the independence number and the dissociation number

The independence number α ( G ) $\alpha (G)$ and the dissociation diss $\text{diss}(G)$ of a graph $G$ are largest orders induced subgraphs maximum degree at most 0 1, respectively. We consider possible improvements obvious inequality 2 ≥ $2\alpha (G)\ge \text{diss}(G)$ . For connected cubic graphs distinct from K 4 ${K}_{4}$ , we show 5 3 $5\alpha 3\,\text{diss}(G)$ describe rich interesting structure extremal in detail. bipartite graphs, and, more generally, triangle-free also obtain improvements. subcubic though, cannot be improved general, characterize all graphs.