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A matching M in a graph G is uniquely restricted if there is no matching in G that is distinct from M but covers the same vertices as M. Solving a problem posed by Golumbic, Hirst, and Lewenstein, we characterize the graphs in which some maximum matching is uniquely restricted. Solving a problem posed by Levit and Mandrescu, we characterize the graphs in which every maximum matching is uniquely...
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...
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