Graph vulnerability parameters, compression, and quasi-threshold graphs

Kelmans, and later independently Bogdanowicz and Satyanarayana, Schoppmann, and Suffel, showed that a graph operation which has come to be known as the compression of G from vertex u to vertex v could not increase, and typically decreased, both the number of spanning trees and the all-terminal reliability of a graph. Both these quantities are well-known vulnerability parameters, i.e., measures of the strength of a network, and subsequently a number of other prominent vulnerability parameters — including vertex connectivity, toughness, scattering number, edge connectivity, edge toughness, and binding number — have been shown to be affected by compression in a similar way. As a consequence threshold graphs are extremal for all of the parameters mentioned. In this paper we show that for the graph vulnerability parameters integrity, tenacity, and k-component order connectivity, if u,v are adjacent then compression cannot increase, and typically decreases them. As a consequence, these parameters have quasi-threshold graphs as extremal graphs. We also show, however, that there are graphs with non-adjacent u,v where compression increases these parameters. To the best of our knowledge, these parameters are the first identified that behave differently under compression depending upon which pairs of vertices are used in the compression.