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Structural Characterisations of (n-1,n)-Trees

Gaurav Kottari, Niteesh Sahni, Qazi J. Azhad

Abstract

We study higher-dimensional analogues of graph-theoretic trees within the class of pure n-simplicial complexes. Focusing on the case m = n-1 in Dewdney's (m, n)-tree framework, we introduce refined notions of path and circuit sequences that overcome the structural limitations of existing definitions. Using these refinements, we establish higher-dimensional analogues of the classical characterisations of trees in graphs, including equivalences based on connectivity, acyclicity, path uniqueness, and enumerative constraints. We further disprove two conjectures posed by Dewdney by constructing explicit counterexamples, and we formulate corrected versions that hold under additional necessary conditions in the case m = n-1. These results provide a structurally complete theory of (n-1, n)-trees, parallel to the classical theory of graph-theoretic trees.

Structural Characterisations of (n-1,n)-Trees

Abstract

We study higher-dimensional analogues of graph-theoretic trees within the class of pure n-simplicial complexes. Focusing on the case m = n-1 in Dewdney's (m, n)-tree framework, we introduce refined notions of path and circuit sequences that overcome the structural limitations of existing definitions. Using these refinements, we establish higher-dimensional analogues of the classical characterisations of trees in graphs, including equivalences based on connectivity, acyclicity, path uniqueness, and enumerative constraints. We further disprove two conjectures posed by Dewdney by constructing explicit counterexamples, and we formulate corrected versions that hold under additional necessary conditions in the case m = n-1. These results provide a structurally complete theory of (n-1, n)-trees, parallel to the classical theory of graph-theoretic trees.

Paper Structure

This paper contains 9 sections, 17 theorems, 84 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial complexes
  4. Dewdney’s framework
  5. Paths, cycles, and connectivity in pure simplicial complexes
  6. Connectivity and components
  7. Reduced simplicial paths
  8. Simplicial cycles
  9. Characterisations of $(n-1,n)$-trees using paths, cycles, and enumeration

Key Result

Theorem 1.1

A pure $n$-simplicial complex $K$ is an $(m,n)$-tree if and only if the following properties hold for $K$:

Figures (4)

  • Figure 1: Example of a pure 2-simplicial complex $K$ and its components
  • Figure 2: Connected pure $n$-simplicial complex
  • Figure 3: A pure $2$-simplicial complex with no $(1,2)$-circuit sequences that is disconnected, and hence not an $(1,2)$-tree.
  • Figure 4: Disconnected pure $n$-simplicial complex with six vertices, four 2-simplices, and contains no $(1,2)$-circuit sequence

Theorems & Definitions (44)

  • Theorem 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 34 more