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For an integer ℓ at least 3, we prove that if G is a graph containing no two vertex‐disjoint circuits of length at least ℓ, then there is a set X of at most vertices that intersects all circuits of length at least ℓ. Our result improves the bound due to Birmelé, Bondy, and Reed (The Erdős–Pósa property for long circuits, Combinatorica 27 (2007), 135–145) who conjecture that ℓ vertices...
The inequality ρ(G)≤γ(G) between the packing number ρ(G) and the domination number γ(G) of a graph G is well known. For general graphs G, there exists no upper bound on γ(G) of the form γ(G)≤f(ρ(G)) where f is a function, as is remarked in [Discrete Math. 309 (2009), 2473–2478]. In this paper, we observe that γ(G)≤Δ(G)ρ(G), where Δ(G) denotes the maximum degree of G. We characterize connected graph...
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