Given a connected graph G on the vertex set \(\{0,1,\ldots ,n\}\) with the root vertex 0, Postnikov and Shapiro associated a monomial ideal \(\mathcal {M}_G\) in the polynomial ring \(R=\mathbb {K}[x_1,\ldots ,x_n]\) over a field \(\mathbb {K}\) such that the vector space dimension of \(R/\mathcal {M}_G\) is same as the determinant of the reduced Laplacian of G. Dochtermann introduced the 1-skeleton ideal \(\mathcal {M}_G^{(1)}\subset \mathcal {M}_G\) which satisfies the property that the vector space dimension of \(R/\mathcal {M}_G^{(1)}\) is bounded below by the determinant of the reduced signless Laplacian of G. In this paper, we characterize all subgraphs of the complete multigraph \(K_{n+1}^{a,1}\) , in particular all simple graphs G, such that the vector space dimension of \(R/\mathcal {M}_G^{(1)}\) is same as the determinant of the reduced signless Laplacian.
