Vertex Decomposability of the Stanley–Reisner Complex of a Path Ideal

The t-path ideal \(I_t(G)\) $I_t(G)$ 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 \(I_2(G)\) $I_2(G)$ of an (undirected) tree G is vertex decomposable. As a consequence, we show that the Alexander dual \(I_2(G)^{\vee }\) $I_2{(G)}^{\vee }$ of \(I_2(G)\) $I_2(G)$ has linear quotients. For each \(t \ge 3\) $t\ge 3$ , we provide a counterexample of a tree for which the Stanley–Reisner complex of \(I_t(G)\) $I_t(G)$ is not vertex decomposable.