Graph Sensitivity under Join and Decomposition

The sensitivity, $\sigma(G)$, of a finite undirected simple graph $G$ is the smallest maximum degree of an induced subgraph on more than the maximum number of independent vertices. Call an indexed family of graphs $G_n$ with maximum degree $\Delta(G_n) \to \infty$ as $n \to \infty$ sensitive if $\sigma(G_n) \to \infty$, and insensitive otherwise. We describe sensitivity under the join operation and decomposition into stable blocks and construct sensitive and insensitive, primarily non-regular, graph families. We determine the sensitivity explicitly for numerous singly- and doubly-indexed graph families, including certain generalized joins - e.g., complete multipartite graphs and some generalized windmill graphs; general rooted products; and families of corona graphs.