A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γt(G). The graph G is 3t-critical if γt(G)=3 and γt(G+e)=2 for every edge e in the complement of G. We show that no bipartite graph is 3t-critical. The tripartite 3 t-critical graphs are characterized. For every k<3, we prove that there are only a finite number of 3t-critical k-partite graphs. We show that the 5-cycle is the only 3t-critical K3-free graph and that there are only a finite number of 3t-critical K4-free graphs.
| Identifer | oai:union.ndltd.org:ETSU/oai:dc.etsu.edu:etsu-works-17608 |
| Date | 06 July 2011 |
| Creators | Haynes, Teresa W., Henning, Michael A., Van Der Merwe, Lucas C., Yeo, Anders |
| Publisher | Digital Commons @ East Tennessee State University |
| Source Sets | East Tennessee State University |
| Detected Language | English |
| Type | text |
| Source | ETSU Faculty Works |
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