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Factorisation of Greedoid Polynomials of Rooted Digraphs

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Abstract

Gordon and McMahon defined a two-variable greedoid polynomial f(G; t, z) for any greedoid G. They studied greedoid polynomials for greedoids associated with rooted graphs and rooted digraphs. They proved that greedoid polynomials of rooted digraphs have the multiplicative direct sum property. In addition, these polynomials are divisible by \(1 + z\) under certain conditions. We compute the greedoid polynomials for all rooted digraphs up to order six. A polynomial is said to factorise if it has a non-constant factor of lower degree. We study the factorability of greedoid polynomials of rooted digraphs, particularly those that are not divisible by \(1 + z\). We give some examples and an infinite family of rooted digraphs that are not direct sums but their greedoid polynomials factorise.

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Notes

  1. From the proverb: All roads lead to Rome.

  2. More combinatorial data can be found at https://users.cecs.anu.edu.au/~bdm/data/.

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Acknowledgements

We thank Gary Gordon and the referee for their useful feedback.

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Authors and Affiliations

  1. Faculty of Information Technology, Monash University, Clayton, VIC, 3800, Australia

    Kai Siong Yow & Graham Farr

  2. School of Information Technology, Faculty of Science Engineering and Built Environment, Deakin University, Geelong, Australia

    Kerri Morgan

  3. Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, UPM, 43400, Serdang, Selangor, Malaysia

    Kai Siong Yow

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  1. Kai Siong Yow
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  2. Kerri Morgan
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  3. Graham Farr
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Correspondence to Kai Siong Yow.

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Kai Siong Yow: (the research, and most of the writing, was done at Monash University Australia as part of the author’s PhD research)

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Yow, K.S., Morgan, K. & Farr, G. Factorisation of Greedoid Polynomials of Rooted Digraphs. Graphs and Combinatorics 37, 2245–2264 (2021). https://doi.org/10.1007/s00373-021-02347-0

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