The k-conversion number of regular graphs

Abstract

Given a graph and a set an irreversible k -threshold conversion process on G is an iterative process wherein, for each St is obtained from by adjoining all vertices that have at least k neighbors in We call the set S0 the seed set of the process, and refer to S0 as an irreversible k-threshold conversion set, or a k-conversion set, of G if for some The k-conversion number is the size of a minimum k-conversion set of G. A set is a decycling set, or feedback vertex set, if and only if is acyclic. It is known that k-conversion sets in -regular graphs coincide with decycling sets. We characterize k-regular graphs having a k-conversion set of size k, discuss properties of -regular graphs having a k-conversion set of size k, and obtain a lower bound for for -regular graphs. We present classes of cubic graphs that attain the bound for and others that exceed it—for example, we construct classes of 3-connected cubic graphs Hm of arbitrary girth that exceed the lower bound for by at least m.