On two conjectures concerning total domination subdivision number in graphs

Rana Khoeilar (), Hossein Karami () and Seyed Mahmoud Sheikholeslami ()
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Rana Khoeilar: Azarbaijan Shahid Madani University
Hossein Karami: Azarbaijan Shahid Madani University
Seyed Mahmoud Sheikholeslami: Azarbaijan Shahid Madani University

Journal of Combinatorial Optimization, 2019, vol. 38, issue 2, No 1, 333-340

Abstract: Abstract A subset S of vertices of a graph G 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 $$\gamma _t(G)$$ γ t ( G ) is the minimum cardinality of a total dominating set of G. The total domination subdivision number $$\mathrm{sd}_{\gamma _t}(G)$$ sd γ 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. In this paper we prove that for any connected graph G of order $$n\ge 3$$ n ≥ 3 , $$\mathrm{sd}_{\gamma _t}(G)\le \gamma _t(G)+1$$ sd γ t ( G ) ≤ γ t ( G ) + 1 and for any connected graph G of order $$n\ge 5$$ n ≥ 5 , $$\mathrm{sd}_{\gamma _t}(G)\le \frac{n+1}{2}$$ sd γ t ( G ) ≤ n + 1 2 , answering two conjectures posed in Favaron et al. (J Comb Optim 20:76–84, 2010a).

Keywords: Matching; Barrier; Total domination number; Total domination subdivision number (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10878-019-00383-4

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