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Three aspects of the MSTCI problem

Manuel Dubinsky, César Massri, Gabriel Taubin

TL;DR

This work analyzes the minimum intersection number of cycle bases induced by spanning trees (the MSTCI problem) on arbitrary connected graphs. It develops two lower-bounding strategies: a proven bound $l_{n,m}=\tfrac{1}{2}\bigl(\tfrac{\mu^2}{n-1}-\mu\bigr)$ based on nonredundant bond distributions, and a tighter conjectural bound $\bar{l}_{n,m}$ derived from $\mu$-regular graphs; the results highlight equidistribution and short cycle-length as structural goals. It then investigates how $\cap(G)$ changes when moving to successors (graphs with one additional edge) and subgraphs, proving several monotonicity properties and presenting rare counterexamples through exhaustive experiments. A final set of experiments attempts to generalize a universal-vertex theorem to broader graph classes, yielding partial insights and posing questions for future work. Overall, the paper provides structural criteria and empirical evidence guiding lower-bound estimation and the interplay between graph density, universal vertices, and the MSTCI value, with potential implications for cycle-structure sparsity and related matrix representations of cycle bases.

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.

Three aspects of the MSTCI problem

TL;DR

This work analyzes the minimum intersection number of cycle bases induced by spanning trees (the MSTCI problem) on arbitrary connected graphs. It develops two lower-bounding strategies: a proven bound based on nonredundant bond distributions, and a tighter conjectural bound derived from -regular graphs; the results highlight equidistribution and short cycle-length as structural goals. It then investigates how changes when moving to successors (graphs with one additional edge) and subgraphs, proving several monotonicity properties and presenting rare counterexamples through exhaustive experiments. A final set of experiments attempts to generalize a universal-vertex theorem to broader graph classes, yielding partial insights and posing questions for future work. Overall, the paper provides structural criteria and empirical evidence guiding lower-bound estimation and the interplay between graph density, universal vertices, and the MSTCI value, with potential implications for cycle-structure sparsity and related matrix representations of cycle bases.

Abstract

Consider a connected graph and let be a spanning tree of . Every edge induces a cycle in . The intersection of two distinct such cycles is the set of edges of 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.
Paper Structure (39 sections, 25 theorems, 62 equations, 4 figures, 3 tables)

This paper contains 39 sections, 25 theorems, 62 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Overview
  4. Notation
  5. Auxiliary lemmas and definitions
  6. Proof
  7. Proof
  8. Proof
  9. Proof
  10. Proof
  11. Proof
  12. Proof
  13. Lower bounds of the intersection number
  14. Overview
  15. Proof of a lower bound
  16. ...and 24 more sections

Key Result

Lemma 2

Let $e \in E'$ be an edge of $T$ and $b_e$ its corresponding bond, then the edges in $b_e - \{e\}$ are cycle-edges that determine tree-cycles that intersect pairwise.

Figures (4)

  • Figure 1: $8$-node connected graphs maximum $\cap(G)$, minimum $\cap(G)$, mean $\cap(G)$ and $l_{n,m}$ interpolating lines (left); mean/standard deviation $r_G$ (right).
  • Figure 2: Comparison of mean/standard deviation of $\bar{r}_G$ (blue) and mean/standard deviation of $r_G$ (red) over a random sample of size 1000 of $9$-node connected graphs.
  • Figure 3: From top left to bottom right, 8-node graphs $G_1$, $G_2$, $G_3$, $G_4$, $G_5$, $G_6$, with intersection number greater than all its successors.
  • Figure 4: 8-node connected graph histogram (left) and counterexample histogram (right).

Theorems & Definitions (32)

  • Definition 1
  • Lemma 2
  • Lemma 3
  • Corollary 4
  • Lemma 5
  • Definition 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • ...and 22 more