Arithmetical structures on graphs with connectivity one

Given a graph G, an arithmetical structure on G is a pair of positive integer vectors (d, r) such that gcd(rv vertical bar v is an element of V(G)) = 1 and (diag(d) - A)r = 0, where A is the adjacency matrix of G. We describe the arithmetical structures on graph G with a cut vertex v in terms of the arithmetical structures on their blocks. More precisely, if G(1), ..., G(s) are the induced subgraphs of G obtained from each of the connected components of G - v by adding the vertex v and their incident edges, then the arithmetical structures on G are in one to one correspondence with the v-rational arithmetical structures on the G(i)'s. A rational arithmetical structure corresponds to an arithmetical structure where some of the integrality conditions are relaxed.