Three aspects of the MSTCI problem

TL;DR

This paper investigates the MSTCI problem, aiming to find spanning trees with minimal cycle intersections, by establishing bounds, comparing graph variants, and generalizing existing results for graphs with a universal vertex.

Contribution

It introduces two lower bounds for the intersection number, compares intersection numbers across similar graphs, and generalizes a recent result for graphs with a universal vertex.

Findings

Established two lower bounds for the intersection number.

Compared intersection numbers of graphs differing by one edge.

Generalized a recent result for graphs with a universal vertex.

Abstract

Consider a connected graph $G$ and let $T$ be a spanning tree of $G$. Every edge $e \in G-T$ induces a cycle in $T \cup \{e\}$. The intersection of two distinct such cycles is the set of edges of $T$ that belong to both cycles. The MSTCI problem consists in finding a spanning tree that has the least number of such non-empty intersections and the instersection number is the number of non-empty intersections of a solution. In this article we consider three aspects of the problem in a general context (i.e. for arbitrary connected graphs). The first presents two lower bounds of the intersection number. The second compares the intersection number of graphs that differ in one edge. The last is an attempt to generalize a recent result for graphs with a universal vertex.