Abstract: A status quo review is made of the counting problem of labeled connected graphs containing k circles. Then a few formulas are introduced for the counting of several special labeled graphs. And finally, still greater effort is focused on the research of the counting problem of the labeled connected graph containing one circle and sharing one common point with k-1 circles of its k circles and the counting problem of labeled connected graph containing only one common point with G0 and S0, and the corresponding counting formula is given.