A Connectivity Constraint Using Bridges

We present a specialised constraint for enforcing graph connectivity. It is assumed that we have a square symmetrical array A of 0/1 constrained integer variables representing potential undirected edges in a simple graph, such that variable A[u, v] corresponds to the undirected edge (u, v). A search process is then at liberty to select and reject edges. The connectivity constraint ensures that no critical edge may be deleted from A, i.e. an edge that if deleted disconnects the graph. This connectivity constraint might then be used when modelling network design problems, modelling molecular structures, problems in bioinformatics [4], or graph problems where connectivity is an essential property [3, 1]. The constraint’s internals are founded on the depth first search (dfs) process. The dfs process generates two sets of edges: the set of tree edges T , and the set of back edges B. The connectivity constraint associates a counter nc[u, v], initialised to zero, with each tree edge. On creation of a back edge (t, w) we increment the counters on the tree edges spanned by that back edge, i.e. we increment the set of counters {nc[u, v] | (u, v) ∈ path(t, w)} where path(t, w) delivers the set of edges that make up the path from t to w in T . A counter is incremented to indicate that there is at least one more cycle in the graph involving edge (u, v), i.e. nc[u, v] is a count of the number of cycles passing through edge (u, v) using edges in T and B. If on termination of the dfs process any counter, say nc[v, w], equals zero then that edge (v, w) is a bridge (also known as an isthmus or cutedge) and must be forced. Consequently the value 0 is removed from the domain of variable A[u, v].