$$(K_{1}\vee {P_{t})}$$ ( K 1 ∨ P t ) -saturated graphs with minimum number of edges

Jinze Hu (), Shengjin Ji () and Qing Cui ()
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Jinze Hu: Shandong University of Technology
Shengjin Ji: Shandong University of Technology
Qing Cui: Nanjing University of Aeronautics and Astronautics

Journal of Combinatorial Optimization, 2025, vol. 49, issue 2, No 6, 15 pages

Abstract: Abstract For a fixed graph F, a graph G is F-saturated if G does not contain F as a subgraph, but adding any edge in $$E(\overline{G})$$ E ( G ¯ ) will result in a copy of F. The minimum size of an F-saturated graph of order n is called the saturation number of F, denoted by sat(n, F). In this paper, we are interested in saturation problem of graph $$K_1\vee {P_t}$$ K 1 ∨ P t for $$t\ge 2$$ t ≥ 2 . As some known results, $$sat(n,K_1\vee {P_t})$$ s a t ( n , K 1 ∨ P t ) is determined for $$2\le t\le 4$$ 2 ≤ t ≤ 4 . We will show that $$sat(n,K_1\vee {P_t})=(n-1)+sat(n-1,P_t)$$ s a t ( n , K 1 ∨ P t ) = ( n - 1 ) + s a t ( n - 1 , P t ) for $$t\ge 5$$ t ≥ 5 and n sufficiently large. Moreover, $$(K_1\vee {P_t})$$ ( K 1 ∨ P t ) -saturated graphs with $$sat(n,K_1\vee {P_t})$$ s a t ( n , K 1 ∨ P t ) edges are characterized.

Keywords: Saturation number; Saturated graphs; Join; 05C75; 05C35 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10878-024-01256-1

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