On Stars in Caterpillars and Lobsters

TL;DR

This paper investigates the structure of maximum stars in specific graph families, proving the Hurlbert and Kamat conjecture for caterpillars and sunlet graphs, and characterizing centers in lobsters.

Contribution

Introduces a new bounding tool for star sizes and applies it to confirm the conjecture for certain graph classes and analyze star centers in lobsters.

Findings

Caterpillars and sunlet graphs satisfy the conjecture.

Largest stars in lobsters are centered at leafs or degree-2 spinal vertices.

Provides bounds on star sizes based on vertex types.

Abstract

The family of all $k$-independent sets of a graph containing a fixed vertex $v$ is called a {star} and $v$ is called its center. Stars are interesting for their relation to Erd\"{o}s-Ko-Rado graphs. Hurlbert and Kamat conjectured that in trees the largest stars are centered in leafs. This conjecture was disproven independently by Baber, Borg, and Feghali, Johnson, and Thomas. In this paper we introduce a tool to bound the size of stars centered at certain vertices by stars centered at leafs. We use this tool to show that caterpillars and sunlet graphs satisfy Hurlbert and Kamat's conjecture, and to show that the centers of the largest stars in lobsters are either leafs or spinal vertices of degree 2.