On the relation between connectivity, independence and generalized caterpillars

TL;DR

This paper explores the relationship between spanning generalized caterpillars, graph connectivity, and independence numbers, correcting a previous theorem and establishing new links in graph theory.

Contribution

It introduces a relation between spanning generalized caterpillars and graph parameters, and clarifies an error in prior literature.

Findings

Established a relation between spanning generalized caterpillars and graph parameters.

Corrected an error in a previous theorem from literature.

Linked another theorem to the existence of spanning generalized caterpillars.

Abstract

A spanning generalized caterpillar is a spanning tree in which all vertices of degree more than two are on a path. In this note, we find a relation between the existence of spanning generalized caterpillar and the independence and connectivity number in a graph. We also point out to an error in a "theorem" in the paper "Spanning spiders and light-splitting switches", by L. Gargano et al. in Discrete Math. (2004), and find out a relation between another mentioned theorem and the existence of spanning generalized caterpillar.