The k-conversion number of regular graphs
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Given a graph G=(V,E) and a set S_0⊆ V, an irreversible k-threshold conversion process on G is an iterative process wherein, for each t=1,2,..., S_t is obtained from S_t-1 by adjoining all vertices that have at least k neighbours in S_t-1. We call the set S_0 the seed set of the process, and refer to S_0 as an irreversible k-threshold conversion set, or a k-conversion set, of G if S_t=V(G) for some t≥ 0. The k-conversion number c_k(G) is the size of a minimum k-conversion set of G. A set X⊆ V is a decycling set, or feedback vertex set, if and only if G[V-X] is acyclic. It is known that k-conversion sets in (k+1)-regular graphs coincide with decycling sets. We characterize k-regular graphs having a k-conversion set of size k, discuss properties of (k+1)-regular graphs having a k-conversion set of size k, and obtain a lower bound for c_k(G) for (k+r)-regular graphs. We present classes of cubic graphs that attain the bound for c_2(G), and others that exceed it---for example, we construct classes of 3-connected cubic graphs H_m of arbitrary girth that exceed the lower bound for c_2(H_m) by at least m.
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