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UPPER BOUNDS ON THE SEMITOTAL FORCING NUMBER OF GRAPHS

Published online by Cambridge University Press:  19 June 2023

YI-PING LIANG*
Affiliation:
School of Mathematics and Statistics, Gansu Center for Applied Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China
JIE CHEN
Affiliation:
School of Mathematics and Statistics, Gansu Center for Applied Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China e-mail: chenjie21@lzu.edu.cn
SHOU-JUN XU
Affiliation:
School of Mathematics and Statistics, Gansu Center for Applied Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PR China e-mail: shjxu@lzu.edu.cn

Abstract

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 $G=K_{\Delta ,\Delta }$.

Type
Research Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by National Natural Science Foundation of China (Grants Nos. 12071194, 11571155).

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