Abstract
A graph is subeulerian if it is spanned by an eulerian supergraph. Boesch, Suffel and Tindell have characterized the class of subeulerian graphs and determined the minimum number of additional lines required to make a subeulerian graph eulerian.In this paper, we consider the related notion of a subsemi-eulerian graph, i.e. a graph which is spanned by a supergraph having an open trail containing all of its lines. The subsemi-eulerian graphs are characterized and formulas for the minimum number of required additional lines are given. Interrelationships between the two problems are stressed as well.