There are two main purposes of this article. First we show that every
3-connected graph embedded in the torus or the Klein bottle has a span
ning planar subgraph which is 2-connected, and in fact has a slightly
stronger connectivity property. Second, this subgraph is applied to sh
ow that every 3-connected graph that embeds in the torus or Klein bott
le has both a 2-walk (a closed walk visiting every vertex exactly once
or twice) and a 3-tree (a spanning tree with maximum degree at most 3
). This completes the characterization of surfaces for which every emb
edded 3-connected graph has a 2-walk (or 3-tree). (C) 1995 Academic Pr
ess, Inc.