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Some results on the total (zero) forcing number of a graph

Jianxi Li (), Dongxin Tu and Wai Chee Shiu ()
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Jianxi Li: Minnan Normal University
Dongxin Tu: Minnan Normal University
Wai Chee Shiu: The Chinese University of Hong Kong

Journal of Combinatorial Optimization, 2025, vol. 49, issue 3, No 1, 22 pages

Abstract: Abstract Let F(G) and $$F_t(G)$$ F t ( G ) be the zero forcing number and the total forcing number of a graph G, respectively. In this paper, we study the relationship between the total forcing number of a graph and its vertex covering number (or independence number), and prove that $$F_t(G) \le \Delta \alpha (G)$$ F t ( G ) ≤ Δ α ( G ) and $$F_t(G) \le (\Delta - 1)\beta (G) + 1$$ F t ( G ) ≤ ( Δ - 1 ) β ( G ) + 1 for any connected graph G with the maximum degree $$\Delta $$ Δ , where $$\alpha (G)$$ α ( G ) and $$\beta (G)$$ β ( G ) are the independence number and the vertex covering number of G. In particular, we prove that $$F_t(T) \le F(T) + \beta (T)$$ F t ( T ) ≤ F ( T ) + β ( T ) for any tree T and characterize all trees T with $$F_t(T) = F(T) + \beta (T)$$ F t ( T ) = F ( T ) + β ( T ) . At the same time, all trees T with $$F_t(T) = (\Delta - 1)\beta (T) + 1$$ F t ( T ) = ( Δ - 1 ) β ( T ) + 1 are completely characterized. In addition, we explore trees, unicycle graphs and Halin graphs satisfying $$F(G) \le \alpha (G)+1$$ F ( G ) ≤ α ( G ) + 1 .

Keywords: Zero forcing number; Total forcing number; Vertex covering number; Vertex independence number; 05C69 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10878-025-01268-5

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