Graph vulnerability parameters, compression, and threshold graphs

Abstract

Given a simple graph G and vertices u,v∈V(G), let NG(u) and NG(v) denote the neighborhoods in G of u and v respectively. The compression of G from u to v produces a new graph Gu→v by, for each x∈NG(u)−NG(v)−{v}, removing edges from G of the form ux and replacing them with corresponding edges of the form vx. Kelmans, and independently Satyanarayana, Schoppmann, and Suffel, showed that for any graph G and any u,v∈V(G), compression from u to v could not increase, and typically decreased, both the number of spanning trees and the all-terminal reliability of G. Both the number of spanning trees and all-terminal reliability are vulnerability parameters, i.e., measures of the strength of a network. We show that a number of other prominent vulnerability parameters—including vertex connectivity, toughness, scattering number, edge connectivity, edge toughness, and binding number—are affected by compression in the same way as number of spanning trees and all-terminal reliability. As a consequence, as with the number of spanning trees and the all-terminal reliability, threshold graphs are extremal graphs for all of the vulnerability parameters considered.