The k-conversion number of regular graphs
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 conver...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Taylor & Francis Group
2020-09-01
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| Series: | AKCE International Journal of Graphs and Combinatorics |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1016/j.akcej.2019.12.016 |
| Summary: | 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. |
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| ISSN: | 0972-8600 2543-3474 |