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On two conjectures concerning total domination subdivision number in graphs

Author

Listed:
  • Rana Khoeilar

    (Azarbaijan Shahid Madani University)

  • Hossein Karami

    (Azarbaijan Shahid Madani University)

  • Seyed Mahmoud Sheikholeslami

    (Azarbaijan Shahid Madani University)

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).

Suggested Citation

  • Rana Khoeilar & Hossein Karami & Seyed Mahmoud Sheikholeslami, 2019. "On two conjectures concerning total domination subdivision number in graphs," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 333-340, August.
    • Handle: RePEc:spr:jcomop:v:38:y:2019:i:2:d:10.1007_s10878-019-00383-4
    DOI: 10.1007/s10878-019-00383-4
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    References listed on IDEAS

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    1. Odile Favaron & Hossein Karami & Seyyed Mahmoud Sheikholeslami, 2011. "Bounding the total domination subdivision number of a graph in terms of its order," Journal of Combinatorial Optimization, Springer, vol. 21(2), pages 209-218, February.
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    1. H. Karami & R. Khoeilar & S. M. Sheikholeslami, 2013. "The total domination subdivision number in graphs with no induced 3-cycle and 5-cycle," Journal of Combinatorial Optimization, Springer, vol. 25(1), pages 91-98, January.

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