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Vertex decomposable complexes of directed forests, conflict graphs and chordality

Priyavrat Deshpande, Rutuja Sawant

TL;DR

The paper studies the simplicial complexes $\,Dlf(D)$ and $\,DT(D)$ arising from directed forests in a multidigraph $D$. By encoding local conflicts among directed edges in graphs (and, when needed, in hypergraphs), the authors realize these complexes as independence complexes $Ind(G_{D}^{lf})$ and $Ind(G_{D}^{t})$, and more generally $Ind(H_{D}^{lf})$ and $Ind(H_{D}^{t})$. This enables applying chordality and $W$-chordality techniques to establish vertex decomposability, shellability, and sequential Cohen–Macaulayness under various acyclicity and structural assumptions, and to describe explicit forbidden configurations that obstruct these properties. The work thus provides a unified combinatorial-topological framework linking directed-forest complexes with independence complexes and hypergraph chordality, and suggests directions for complete classifications, homotopy types, and algorithmic tests.

Abstract

Let $D$ be a multidigraph. We study the simplicial complex $\mathrm{Dlf}(D)$, whose vertices are the directed edges of $D$ and whose faces correspond to directed linear forests, that is, vertex-disjoint unions of directed paths. We also consider the related directed tree complex $\mathrm{DT}(D)$. Our main approach is to associate with $D$ a simple graph encoding the local incompatibilities among the edges of $D$. Under mild acyclicity assumptions, we show that $\mathrm{Dlf}(D)$ and $\mathrm{DT}(D)$ can be realized as the independence complexes of respective graphs. This correspondence allows us to apply structural results from the theory of independence complexes to obtain graph-theoretic criteria guaranteeing vertex decomposability, shellability, and sequential Cohen-Macaulayness of these complexes. In particular, we describe explicit forbidden induced directed subgraphs that obstruct vertex decomposability, and we identify classes of multidigraphs-including certain acyclic multidigraphs and multidigraphs whose underlying graphs are forests or cycles-for which $\mathrm{Dlf}(D)$ and $\mathrm{DT}(D)$ are vertex decomposable. We also provide examples showing that these properties do not hold in general.

Vertex decomposable complexes of directed forests, conflict graphs and chordality

TL;DR

The paper studies the simplicial complexes and arising from directed forests in a multidigraph . By encoding local conflicts among directed edges in graphs (and, when needed, in hypergraphs), the authors realize these complexes as independence complexes and , and more generally and . This enables applying chordality and -chordality techniques to establish vertex decomposability, shellability, and sequential Cohen–Macaulayness under various acyclicity and structural assumptions, and to describe explicit forbidden configurations that obstruct these properties. The work thus provides a unified combinatorial-topological framework linking directed-forest complexes with independence complexes and hypergraph chordality, and suggests directions for complete classifications, homotopy types, and algorithmic tests.

Abstract

Let be a multidigraph. We study the simplicial complex , whose vertices are the directed edges of and whose faces correspond to directed linear forests, that is, vertex-disjoint unions of directed paths. We also consider the related directed tree complex . Our main approach is to associate with a simple graph encoding the local incompatibilities among the edges of . Under mild acyclicity assumptions, we show that and can be realized as the independence complexes of respective graphs. This correspondence allows us to apply structural results from the theory of independence complexes to obtain graph-theoretic criteria guaranteeing vertex decomposability, shellability, and sequential Cohen-Macaulayness of these complexes. In particular, we describe explicit forbidden induced directed subgraphs that obstruct vertex decomposability, and we identify classes of multidigraphs-including certain acyclic multidigraphs and multidigraphs whose underlying graphs are forests or cycles-for which and are vertex decomposable. We also provide examples showing that these properties do not hold in general.
Paper Structure (12 sections, 19 theorems, 37 equations, 8 figures)

This paper contains 12 sections, 19 theorems, 37 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial complexes
  4. Multidigraphs
  5. Associated Complexes
  6. Independence Complex
  7. The complex of Directed Linear Forests
  8. The Complex of Directed Trees
  9. Going beyond pairwise conflicts
  10. Conflict hypergraph for the directed tree complex
  11. Conflict hypergraph for the directed linear forest
  12. Concluding Remarks

Key Result

Lemma 2.1

For the directed linear forests complex, for every $e \in E(D)$ we have:

Figures (8)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 5: $L_n$
  • Figure 6: $P_n$
  • ...and 3 more figures

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.1
  • proof
  • Definition 2.8
  • ...and 50 more