  The weak minor G of a graph G is the graph obtained from G by a sequence of
edge-contraction operations on G. A weak-minor-closed family of upper
embeddable graphs is a set G of upper embeddable graphs that for each graph G
in G, every weak minor of G is also in G. Up to now, there are few results
providing the necessary and sufficient conditions for characterizing upper
embeddability of graphs. In this paper, we studied the relation between the
vertex splitting operation and the upper embeddability of graphs; provided not
only a necessary and sufficient condition for characterizing upper
embeddability of graphs, but also a way to construct weak-minor-closed family
of upper embeddable graphs from the bouquet of circles; extended a result in J:
Graph Theory obtained by L. Nebesk{\P}y. In addition, the algorithm complex of
determining the upper embeddability of a graph can be reduced much by the
results obtained in this paper.
