A graph G is called collapsible if for every even subset R⊆V(G),
there is a spanning connected subgraph H of G such that R is the set of
vertices of odd degree in H. A graph is the reduction of G if it is
obtained from G by contracting all the nontrivial collapsible subgraphs. A
graph is reduced if it has no nontrivial collapsible subgraphs. In this paper,
we first prove a few results on the properties of reduced graphs. As an
application, for 3-edge-connected graphs G of order n with d(u)+d(v)≥2(n/p−1) for any uv∈E(G) where p>0 are given, we show how such graphs
change if they have no spanning Eulerian subgraphs when p is increased from
p=1 to 10 then to 15