Spanning trees with nonseparating paths

TL;DR

This paper investigates spanning trees with the property that removing any path leaves the graph connected, revealing connections to Hamiltonicity in planar and low-connectivity graphs, and exploring nonseparating cycles in the cycle space.

Contribution

It establishes the relationship between such spanning trees and Hamiltonicity in planar and low-connectivity graphs, and examines nonseparating fundamental cycles.

Findings

Existence of such trees is linked to Hamiltonicity in planar graphs.

For graphs with small vertex cuts, Hamiltonicity remains a key factor.

The study explores nonseparating fundamental cycles and bases in the cycle space.

Abstract

We consider questions related to the existence of spanning trees in graphs with the property that after the removal of any path in the tree the graph remains connected. We show that, for planar graphs, the existence of trees with this property is closely related to the Hamiltonicity of the graph. For graphs with a 1- or 2-vertex cut, the Hamiltonicity also plays a central role. We also deal with spanning trees satisfying this property restricted to paths arising from fundamental cycles. The cycle space of a graph can be generated by the fundamental cycles of any spanning tree, and Tutte showed, that for a 3-connected graph, it can be generated by nonseparating cycles. We are also interested in the existence of a fundamental basis consisting of nonseparating cycles.