Packing disjoint cycles over vertex cuts

For a graph G , let ν ( G ) and ν ′ ( G ) denote the maximum cardinalities of packings of vertex-disjoint and edge-disjoint cycles of G , respectively. We study the interplay of these two parameters and vertex cuts in graphs. If G is a graph whose vertex set can be partitioned into three non-empty sets S , V 1 , and V 2 such that there is no edge between V 1 and V 2 , and k = | S | , then our results imply that ν ( G ) is uniquely determined by the values ν ( H ) for at most 2 k + 1 k ! 2 graphs H of order at most max { | V 1 | , | V 2 | } + k , and ν ′ ( G ) is uniquely determined by the values ν ′ ( H ) for at most 2 k 2 + 1 graphs H of order at most max { | V 1 | , | V 2 | } + k .