# Perfect Forests in Graphs and Their Extensions

Let G be a graph on n vertices. For i∈{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.

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