Perfect Forests in Graphs and Their Extensions

Let $G$ be a graph on $n$ vertices. For $i\in {0,1}$ and a connected graph $G$, a spanning forest $F$ of $G$ is called an $i$-perfect forest if every tree in $F$ is an induced subgraph of $G$ and exactly $i$ vertices of $F$ have even degree (including zero). A $i$-perfect forest of $G$ is proper if it has no vertices of degree zero. Scott (2001) showed that every connected graph with even number of vertices contains a (proper) 0-perfect forest. We prove that one can find a 0-perfect forest with minimum number of edges in polynomial time, but it is NP-hard to obtain a 0-perfect forest with maximum number of edges. Moreover, we show that to decide whether $G$ has a 0-perfect forest with at least $|V(G)|/2+k$ edges, where $k$ is the parameter, is W[1]-hard. We also prove that for a prescribed edge $e$ of $G,$ it is NP-hard to obtain a 0-perfect forest containing $e,$ but one can decide if there existsa 0-perfect forest not containing $e$ in polynomial time. It is easy to see that every graph with odd number of vertices has a 1-perfect forest. It is not the case for proper 1-perfect forests. We give a characterization of when a connected graph has a proper 1-perfect forest.