Graph Sensitivity under Join and Decomposition

TL;DR

This paper investigates the concept of graph sensitivity, analyzing how it behaves under join and decomposition operations, and constructs various sensitive and insensitive graph families with explicit sensitivity calculations.

Contribution

It introduces the notion of sensitivity for graphs, describes its behavior under join and decomposition, and constructs explicit sensitive and insensitive graph families.

Findings

Sensitivity can be explicitly determined for various graph families.

Join and decomposition operations significantly affect graph sensitivity.

Constructed non-regular sensitive and insensitive graph families.

Abstract

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.