Perfect Forests in Graphs and Their Extensions

TL;DR

This paper investigates perfect forests in graphs, providing polynomial algorithms for some cases, proving NP-hardness for others, and characterizing conditions for the existence of certain perfect forests.

Contribution

It introduces new complexity results and algorithms for finding perfect forests with specific properties in graphs.

Findings

Polynomial-time algorithm for minimum edge 0-perfect forests.

NP-hardness of maximum edge 0-perfect forests.

W[1]-hardness for certain edge count thresholds.

Abstract

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. 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.