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UPPER BOUNDS ON THE SEMITOTAL FORCING NUMBER OF GRAPHS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 2 / April 2024
- Published online by Cambridge University Press:
- 19 June 2023, pp. 177-185
- Print publication:
- April 2024
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- Article
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Let G be a graph with no isolated vertex. A semitotal forcing set of G is a (zero) forcing set S such that every vertex in S is within distance 2 of another vertex of S. The semitotal forcing number
$F_{t2}(G)$ is the minimum cardinality of a semitotal forcing set in G. In this paper, we prove that it is NP-complete to determine the semitotal forcing number of a graph. We also prove that if 
$G\neq K_n$ is a connected graph of order 
$n\geq 4$ with maximum degree 
$\Delta \geq 2$, then 
$F_{t2}(G)\leq (\Delta-1)n/\Delta$, with equality if and only if either 
$G=C_{4}$ or 
$G=P_{4}$ or 
$G=K_{\Delta ,\Delta }$.
