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3 has total domination number at most its matching number. In general, we show that no minimum degree is sufficient to guarantee that the matching number and total domination number are comparable.
doi:10.1016/j.disc.2004.06.004 
Copyright © 2004 Elsevier B.V. All rights reserved.
Total domination subdivision numbers of trees
Received 1 May 2003;
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Abstract
A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. The total domination number γt(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt(G) of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. Haynes et al. (J. Combin. Math. Combin. Comput. 44 (2003) 115) showed that for any tree T of order at least 3, 1
sdγt(T)3. In this paper, we give a constructive characterization of trees whose total domination subdivision number is 3.
Keywords: Trees; Total domination number; Total domination subdivision number
MSC: 05C69
