Abstract
AbstractLet be a graph on vertices. For , a spanning forest of is called an ‐perfect forest if every tree in is an induced subgraph of and exactly vertices of have even degree (including zero). An ‐perfect forest of is proper if it has no vertices of degree zero. Scott showed that every connected graph with an even number of vertices contains a (proper) 0‐perfect forest. We prove that one can find a 0‐perfect forest with a minimum number of edges in polynomial time, but it is NP‐hard to obtain a 0‐perfect forest with a maximum number of edges. Moreover, we show that to decide whether has a 0‐perfect forest with at least edges, where is the parameter, is W[1]‐hard. We also prove that for a prescribed edge of it is NP‐hard to obtain a 0‐perfect forest containing but one can decide if there exists a 0‐perfect forest not containing in polynomial time. It is easy to see that every connected graph with an 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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