Vertex decomposable complexes of directed forests, conflict graphs and chordality

TL;DR

This paper investigates the combinatorial and topological properties of complexes derived from directed forests and multidigraphs, establishing conditions for vertex decomposability and related properties using graph-theoretic methods.

Contribution

It introduces a novel approach linking directed forest complexes to independence complexes of graphs, providing new criteria for their structural properties.

Findings

Vertex decomposability is guaranteed under certain acyclicity conditions.

Explicit forbidden subgraphs obstructing vertex decomposability are characterized.

Classes of multidigraphs with forests or cycles have vertex decomposable complexes.

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.