Cycles through given vertices and closures Original Research Article

A set S⊆V(G) is cyclable in G if G contains a cycle C with S⊆V(C), and pancyclable in G if G contains cycles Ci with |S∩V(Ci)|=i for all i, 3⩽i⩽|S|. We consider stability of the properties of cyclability and pancylability of a given set S under the Bondy–Chvátal closure and under the ∗-closure introduced by the first author. We give refinements of the closure concepts by localizing “close” to the set S and we also derive some sufficient degree conditions for cyclability (of Ore type with d(x)+d(y)⩾n−1 and with σ3⩾n−2 restricted to S).