# Decycling Number of Linear Graphs of Trees

## TL;DR

This paper investigates the decycling number of line graphs of trees, providing bounds and exact values for specific classes, thereby advancing understanding of acyclic subgraph removal in these graph structures.

## Contribution

It establishes bounds on the decycling number of line graphs of trees and determines exact values for perfect k-ary trees, introducing extremal constructions.

## Key findings

- Bounds on decycling number for line graphs of trees
- Exact decycling number for line graphs of perfect k-ary trees
- Extremal trees achieving bounds

## Abstract

The decycling number of a graph $G$ is the minimum number of vertices whose removal from $G$ results in an acyclic subgraph. It is known that determining the decycling number of a graph $G$ is equivalent to finding the maximum induced forests of $G$. The line graphs of trees are the claw-free block graphs. These graphs have been used by Erd\H{o}s, Saks and S\'{o}s to construct graphs with a given number of edges and vertices whose maximum induced tree is very small. In this paper, we give bounds on the decycling number of line graphs of trees and construct extremal trees to show that these bounds are the best possible. We also give bounds on the decycling number of line graph of $k$-ary trees and determine the exact the decycling number of line graphs of perfect $k$-ary trees.