Given a connected graph G on the vertex set {0,1,…,n} with the root vertex 0, Postnikov and Shapiro associated a monomial ideal MG in the polynomial ring R=K[x1,…,xn] over a field K such that the vector space dimension of R/MG is same as the determinant of the reduced Laplacian of G. Dochtermann introduced the 1-skeleton ideal MG(1)⊂MG which satisfies the property that the vector space dimension of R/MG(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 Kn+1a,1, in particular all simple graphs G, such that the vector space dimension of R/MG(1) is same as the determinant of the reduced signless Laplacian.