  We consider problems of finding a maximum size/weight $t$-matching without
forbidden subgraphs in an undirected graph $G$ with the maximum degree bounded
by $t+1$, where $t$ is an integer greater than $2$. Depending on the variant
forbidden subgraphs denote certain subsets of $t$-regular complete partite
subgraphs of $G$. A graph is complete partite if there exists a partition of
its vertex set such that every pair of vertices from different sets is
connected by an edge and vertices from the same set form an independent set. A
clique $K_t$ and a bipartite clique $K_{t,t}$ are examples of complete partite
graphs. These problems are natural generalizations of the triangle-free and
square-free $2$-matching problems in subcubic graphs. In the weighted setting
we assume that the weights of edges of $G$ are vertex-induced on every
forbidden subgraph. We present simple and fast combinatorial algorithms for
these problems. The presented algorithms are the first ones for the weighted
versions, and for the unweighted ones, are faster than those known previously.
Our approach relies on the use of gadgets with so-called half-edges. A
half-edge of edge $e$ is, informally speaking, a half of $e$ containing exactly
one of its endpoints.
