A 2-matching complex is a simplicial complex which captures the
relationship between 2-matchings of a graph. In this paper, we will use
discrete Morse Theory and the Matching Tree Algorithm to prove homotopical
results. We will consider a class of graphs for which the homotopy type of the
2-matching complex transforms from a sphere to a point with the addition of
leaves. We end the paper by defining k-matching sequences and looking at the
1- and 2-matching complexes of wheel graphs and perfect caterpillar graphs