Independent Sets and Balanced Cycle-Linkings in 2-Connected Graphs

For every $\alpha \ge 3$ and all sufficiently large $n$, we identify a single graph that simultaneously maximizes the number of independent sets of every size among all $n$-vertex $2$-connected graphs with independence number $\alpha$. This graph is unique up to isomorphism and is a balanced cycle-linking of the disjoint union of $\alpha$ cliques whose orders differ by at most one. More precisely, for each $3\le \beta \le \alpha$, the graphs maximizing the number of independent $\beta$-sets are exactly the cycle-linkings whose clique-size cyclic words are $\lfloor \beta \mathrm{/}2\rfloor$-balanced. These results extend the corresponding extremal result for connected graphs to the $2$-connected setting. The proof combines generalized Turán-type clique counting and the edge-extremal theory of $2$-connected graphs with a coefficientwise balancing-switch argument. The switch also shows that a shortest imbalance of length $r$ first affects the independent-set count in degree $2r$.