Abstract
The t-path ideal \(I_t(G)\) of a graph G is the square-free monomial ideal generated by the monomials which correspond to the paths of length t in G. In this paper, we prove that the Stanley–Reisner complex of the 2-path ideal \(I_2(G)\) of an (undirected) tree G is vertex decomposable. As a consequence, we show that the Alexander dual \(I_2(G)^{\vee }\) of \(I_2(G)\) has linear quotients. For each \(t \ge 3\), we provide a counterexample of a tree for which the Stanley–Reisner complex of \(I_t(G)\) is not vertex decomposable.
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Acknowledgements
The author is grateful to his supervisor, Ajay Kumar, and to Rajiv Kumar for their valuable contributions to the discussions regarding the materials covered in this paper. The author made extensive use of the commutative algebra software, Macaulay 2, [6]. The author acknowledges the financial support from CSIR-UGC, New Delhi, India.
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Communicated by Siamak Yassemi.
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Bijender Vertex Decomposability of the Stanley–Reisner Complex of a Path Ideal. Bull. Malays. Math. Sci. Soc. 47, 105 (2024). https://doi.org/10.1007/s40840-024-01699-z
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DOI: https://doi.org/10.1007/s40840-024-01699-z