Graph Sensitivity under Join and Decomposition

TL;DR

This work formalizes graph sensitivity $\sigma(G)$ as the minimum maximum degree over induced subgraphs on $\alpha(G)+1$ vertices and investigates its behavior under the join operation and stable-block decompositions. By deriving explicit formulas for $\sigma(G_1 \vee G_2)$ in terms of $\alpha$-values and $k$-sensitivities $\sigma_k$, the authors obtain concrete results for a wide range of joins, including complete bipartite/multipartite graphs and generalized windmills, as well as for $n$-cones $\overline{K}_n \vee G$. They further develop the stable-block framework to reduce and sometimes exactly determine $\sigma(G)$ when $G$ can be decomposed into stable blocks, and apply these ideas to rooted products and coronas, yielding sharp reductions such as $\sigma(G \odot H) = \sigma(H)$ when $H\neq\varnothing$ and $\sigma(G \odot H)=|V(H)|$ when $H=\varnothing$. The paper thus provides systematic constructions of sensitive and insensitive graph families and raises open questions about generalized joins and modular decompositions, with potential implications for network modeling and related graph invariants. Key results include the exact $\sigma$-values for many joins, a simple formula for regular graphs in $n$-cones, and a decomposition principle that ties sensitivity to stable-block interiors.

Abstract

The sensitivity, $σ(G)$, 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 $Δ(G_n) \to \infty$ as $n \to \infty$ sensitive if $σ(G_n) \to \infty$, 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.

Figures (1)

• Figure 1: Summary of values for sensitivity of $\overline{K}_n \vee G$ in \ref{['n-cone gensensitivity']}. (If $\sigma(G)<\frac{\alpha(G)+1}{2}$ then $\alpha(G)+1-\sigma(G)$ and $\left\lfloor\frac{\alpha(G)+1}{2}\right\rfloor$ are interchanged.)

Theorems & Definitions (70)

• Definition 2.1
• Definition 2.2
• Example 2.3
• Example 2.4
• Lemma 2.5
• proof
• Example 2.6
• Definition 2.7
• Example 2.8
• Example 2.9