Note on Perfect Forests

A spanning subgraph $F$ of a graph $G$ is called perfect if $F$ is a forest, the degree $d_F(x)$ of each vertex $x$ in $F$ is odd, and each tree of $F$ is an induced subgraph of $G$. We provide a short proof of the following theorem of A.D. Scott (Graphs & Combin., 2001): a connected graph $G$ contains a perfect forest if and only if $G$ has an even number of vertices.

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