  The problem of finding a maximum $2$-matching without short cycles has
received significant attention due to its relevance to the Hamilton cycle
problem. This problem is generalized to finding a maximum $t$-matching which
excludes specified complete $t$-partite subgraphs, where $t$ is a fixed
positive integer. The polynomial solvability of this generalized problem
remains an open question. In this paper, we present polynomial-time algorithms
for the following two cases of this problem: in the first case the forbidden
complete $t$-partite subgraphs are edge-disjoint; and in the second case the
maximum degree of the input graph is at most $2t-1$. Our result for the first
case extends the previous work of Nam (1994) showing the polynomial solvability
of the problem of finding a maximum $2$-matching without cycles of length four,
where the cycles of length four are vertex-disjoint. The second result expands
upon the works of B\'{e}rczi and V\'{e}gh (2010) and Kobayashi and Yin (2012),
which focused on graphs with maximum degree at most $t+1$. Our algorithms are
obtained from exploiting the discrete structure of restricted $t$-matchings and
employing an algorithm for the Boolean edge-CSP.
