Note on Perfect Forests in Digraphs

TL;DR

This paper explores generalizations of perfect forests in directed graphs, proving some are computationally feasible and extending Scott's theorem to digraphs.

Contribution

It introduces four generalizations of perfect forests in digraphs, showing three are polynomial-time solvable and extending Scott's theorem.

Findings

Three generalizations are polynomial-time solvable.

Existence of the most straightforward generalization is NP-hard.

Every digraph with one strong component contains these directed forests.

Abstract

A spanning subgraph $F$ of a graph $G$ is called {\em 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 $G$. Alex Scott (Graphs \& Combin., 2001) proved that every connected graph $G$ contains a perfect forest if and only if $G$ has an even number of vertices. We consider four generalizations to directed graphs of the concept of a perfect forest. While the problem of existence of the most straightforward one is NP-hard, for the three others this problem is polynomial-time solvable. Moreover, every digraph with only one strong component contains a directed forest of each of these three generalization types. One of our results extends Scott's theorem to digraphs in a non-trivial way.