  As a fundamental research object, the minimum edge dominating set (MEDS)
problem is of both theoretical and practical interest. However, determining the
size of a MEDS and the number of all MEDSs in a general graph is NP-hard, and
it thus makes sense to find special graphs for which the MEDS problem can be
exactly solved. In this paper, we study analytically the MEDS problem in the
pseudofractal scale-free web and the Sierpi\'nski gasket with the same number
of vertices and edges. For both graphs, we obtain exact expressions for the
edge domination number, as well as recursive solutions to the number of
distinct MEDSs. In the pseudofractal scale-free web, the edge domination number
is one-ninth of the number of edges, which is three-fifths of the edge
domination number of the Sierpi\'nski gasket. Moreover, the number of all MEDSs
in the pseudofractal scale-free web is also less than that corresponding to the
Sierpi\'nski gasket. We argue that the difference of the size and number of
MEDSs between the two studied graphs lies in the scale-free topology.
