Supereulerian index is stable under contractions and closures

The supereulerian index of a graph G is the smallest integer k such that the k-th iterated line graph of G is supereulerian. We first show that adding an edge between two vertices with degree sums at least three in a graph cannot increase its supereulerian index. We use this result to prove that the supereulerian index of a graph G will not be changed after either of contracting an A_G(P)-contractible subgraph F of a graph G and performing the closure operation on G (if G is claw-free). Our results extend a Catlin's remarkable theorem [4] relating that the supereulericity of a graph is stable under the contraction of a collapsible subgraph.