A Gallai’s Theorem type result for the edge stability of graphs

For an arbitrary invariant ρ ( G ) of a graph G the ρ -edge stability number es ρ ( G ) of G is the minimum number of edges of G whose removal results in a graph H ⊆ G with ρ ( H ) (cid:54) = ρ ( G ) . If such an edge set does not exist, then es ρ ( G ) = ∞ . Gallai’s Theorem states that α (cid:48) ( G ) + β (cid:48) ( G ) = n ( G ) for a graph G without isolated vertices, where α (cid:48) ( G ) is the matching number, β (cid:48) ( G ) the edge covering number, and n ( G ) the order of G . We prove a corresponding result for invariants that are based on the edge stability number es ρ ( G )