Vertex decomposable complexes of directed forests, conflict graphs and chordality
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| Format: | Preprint |
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2026
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| author | Deshpande, Priyavrat Sawant, Rutuja |
| author_facet | Deshpande, Priyavrat Sawant, Rutuja |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_14936 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Vertex decomposable complexes of directed forests, conflict graphs and chordality Deshpande, Priyavrat Sawant, Rutuja Combinatorics Primary 05E45, Secondary 05C20, 05C76, 55U10 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. |
| title | Vertex decomposable complexes of directed forests, conflict graphs and chordality |
| topic | Combinatorics Primary 05E45, Secondary 05C20, 05C76, 55U10 |
| url | https://arxiv.org/abs/2602.14936 |