On specific factors in graphs

It is well known that if $$G = (V, E)$$ is a connected multigraph and $$X\subset V$$ is a subset of even order, then G contains a spanning forest H such that each vertex from X has an odd degree in H and all the other vertices have an even degree in H. This spanning forest may have isolated vertices. If this is not allowed in H, then the situation is much more complicated. In this paper, we study this problem and generalize the concepts of even-factors and odd-factors in a unified form.