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EDGE WEIGHTING FUNCTIONS ON THE SEMITOTAL DOMINATING SET OF CLAW-FREE GRAPHS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 2 / October 2024
- Published online by Cambridge University Press:
- 12 February 2024, pp. 177-188
- Print publication:
- October 2024
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- Article
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In an isolate-free graph G, a subset S of vertices is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number of G, denoted by
$\gamma _{t2}(G)$, is the minimum cardinality of a semitotal dominating set in G. Using edge weighting functions on semitotal dominating sets, we prove that if 
$G\neq N_2$ is a connected claw-free graph of order 
$n\geq 6$ with minimum degree 
$\delta (G)\geq 3$, then 
$\gamma _{t2}(G)\leq \frac{4}{11}n$ and this bound is sharp, disproving the conjecture proposed by Zhu et al. [‘Semitotal domination in claw-free cubic graphs’, Graphs Combin. 33(5) (2017), 1119–1130].
