There is a paper by Babenko and Gusakov ("New Exact and Approximation Algorithms for the Star Packing Problem in Undirected Graphs"). It discusses the star packing problem. Given an integer $T > 1$, a star is a subgraph $K_{1, t}$ for some $1 \leq t \leq T$. They suggest a polynomial time algorithm for packing vertex-disjoint $T$-stars into an undirected graph maximizing the number of taken vertices. For $T = 2$ the problem is equivalent to packing connected edges and triplets.
Here is the core idea of the algorithm. For each undirected edge $(u, v)$ we introduce two arcs $(u, v)$ and $(v, u)$. Let the original graph be $G$ and the directed graph be $G'$.
Let us find a subgraph in $G'$ such that for each vertex $v$, $outdeg(v) \leq 1$ and $indeg(v) \leq T$ and maximizing the number of edges. It can be done with the maximum flow algorithm over the following network:
- split each vertex $v$ into $v_{in}$ and $v_{out}$
- for each arc $(u, v)$ add an arc $(u_{out}, v_{in})$ with unit capacity to the network
- for each vertex $v$, add arcs $(Source, v_{out})$ with unit capacity and $(v_{in}, Sink)$ with capacity $T$.
Claim: given such subgraph of $G'$ with $k$ edges, one can build a star packing in $G$ with $k$ vertices, and vice versa.
Proof ($\leftarrow$): for each star with middle vertex $v$ and leaves $u_1, \dots, u_k$ pick arcs $(v, u_1), (u_1, v), (u_2, v), \dots, (u_k, v)$.
Proof sketch ($\rightarrow$): each (weakly) connected component of the subgraph in $G'$ is a functional graph, i.e. a cycle where a rooted tree can be connected to each of its vertices. Traverse vertices of those trees in ordered by depth, decreasing. Whenever a vertex $v$ is traversed, take its parent $p(v)$ and all its siblings, form a star with $p(v)$ in the middle and remove all taken vertices from the tree. Some casework is necessary when the cycle is reached, it is left as an exercise to the reader.
