The total domination subdivision number in graphs with no induced 3-cycle and 5-cycle

H. Karami, R. Khoeilar and S. M. Sheikholeslami ()
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H. Karami: Azarbaijan University of Tarbiat Moallem
R. Khoeilar: Azarbaijan University of Tarbiat Moallem
S. M. Sheikholeslami: Azarbaijan University of Tarbiat Moallem

Journal of Combinatorial Optimization, 2013, vol. 25, issue 1, No 6, 98 pages

Abstract: Abstract A set S of vertices of a graph G=(V,E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γ t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number $\mathrm{sd}_{\gamma_{t}}(G)$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Favaron, Karami, Khoeilar and Sheikholeslami (J. Comb. Optim. 20:76–84, 2010a) conjectured that: For any connected graph G of order n≥3, $\mathrm{sd}_{\gamma_{t}}(G)\le \gamma_{t}(G)+1$ . In this paper we use matching to prove this conjecture for graphs with no 3-cycle and 5-cycle. In particular this proves the conjecture for bipartite graphs.

Keywords: Matching; Barrier; Total domination number; Total domination subdivision number (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10878-011-9421-3

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