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. We also prove that for a prescribed edge $e$ of $G,$ it is NP-hard to obtain a 0-perfect forest containing $e,$ but we can find a 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.
翻译:$G$ 是一个以美元为顶点的图表。 对于 $@ 0. 1 $ 美元 和一个关联的图表 $ G$ 来说, 如果每棵以美元为单位的树都是以美元为单位的子图, 而每棵以美元为单位的树都是以美元为单位(包括零), 美元为美元为单位的顶点, 美元为美元。 美元为美元为单位的完美森林是合适的。 斯科特(2001年) 显示, 每一个与偶数的顶点相联的图都含有( 丙) 零 perfect 的森林。 我们证明, 在多元时间里, 每棵完美的森林可以找到一个零perfectforest的森林, 最少的边点数是$, 但是很难找到一个有最高边点数的顶点的森林。 我们还证明, 对于一个规定的边点为$的边点, 美元为以美元为单位的顶点的森林, 很难获得一个零perperforforal $, 但我们可以找到一个零per perfect freformain is a crefect of frefect of per frium a creful is a crefrum is a crefrum is a crefrus.