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Minimum Spanning Tree Cycle Intersection Problem

Manuel Dubinsky, César Massri, Gabriel Taubin

TL;DR

The paper introduces the Minimum Spanning Tree Cycle Intersection (MSTCI) problem, framing it within cycle-basis theory and motivating an application to mesh deformation. It provides a complete analysis for complete graphs, proving that star spanning trees uniquely minimize the total cycle-intersection number, with a precise closed-form for star-tree intersections and a clear construction achieving the minimum. Extending to graphs that admit a star spanning tree, the authors derive closed formulas for star trees, show local optimality in the spanning-tree graph, and propose a conjecture that cap(T_s) ≤ cap(T) for all spanning trees T, along with reduction techniques to simplify potential counterexamples. Complementing the theoretical results, they conduct programmatic explorations (exhaustive for small graphs and random for larger graphs) that yield no counterexamples, supporting the conjecture and highlighting that MSTCI minimizers depend on embedding rather than intrinsic properties of the spanning trees.

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. We consider the problem of finding a spanning tree that has the least number of such non-empty intersections.

Minimum Spanning Tree Cycle Intersection Problem

TL;DR

The paper introduces the Minimum Spanning Tree Cycle Intersection (MSTCI) problem, framing it within cycle-basis theory and motivating an application to mesh deformation. It provides a complete analysis for complete graphs, proving that star spanning trees uniquely minimize the total cycle-intersection number, with a precise closed-form for star-tree intersections and a clear construction achieving the minimum. Extending to graphs that admit a star spanning tree, the authors derive closed formulas for star trees, show local optimality in the spanning-tree graph, and propose a conjecture that cap(T_s) ≤ cap(T) for all spanning trees T, along with reduction techniques to simplify potential counterexamples. Complementing the theoretical results, they conduct programmatic explorations (exhaustive for small graphs and random for larger graphs) that yield no counterexamples, supporting the conjecture and highlighting that MSTCI minimizers depend on embedding rather than intrinsic properties of the spanning trees.

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. We consider the problem of finding a spanning tree that has the least number of such non-empty intersections.

Paper Structure

This paper contains 29 sections, 14 theorems, 57 equations, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Cycle bases
  3. An application
  4. Preliminaries
  5. Overview
  6. Notation
  7. Closest point
  8. Tree cycle intersection partition
  9. Important remark
  10. Tree cycles of complete graphs
  11. Overview
  12. Cycle intersection number formula
  13. Main result
  14. Further generalization
  15. Overview
  16. ...and 14 more sections

Key Result

Lemma 1

Let $G=(V,E)$ be a connected graph, $T$ a spanning tree of $G$ and $c \in C_T$ a tree-cycle. Then for every node $v \in V$ there exists a unique node $w \in c$ such that Proof. The proof proceeds by contradiction. If $v \in c$, it is its own unique closest point. Suppose that $v \notin c$ and that there are two distinct nodes $w, w' \in c$ such that $|vTw| = |vTw'| \leq |vTu| \ \forall u \in c$,

Theorems & Definitions (21)

  • Lemma 1
  • Definition 2
  • Definition 3
  • Lemma 4
  • Lemma 5
  • Proposition 6
  • Corollary 7
  • Corollary 8
  • Definition 9
  • Definition 10
  • ...and 11 more