Enumeration of connected spanning subgraphs of a planar graph

Abstract

One of the important solved problems in graph theory is the enumeration of distinct connected spanning trees contained in a given connected graph; historically it was first solved by Kirchhoff. Since a tree graph contains no cycle by definition, the next question of importance is naturally the problem of enumeration, for a given connected graph, of connected spanning subgraphs each of which contains only a single cycle (besides the trees attached to the particular cycle). In general, the problems of interest are the enumeration of the connected spanning subgraphs having a preassigned cyclomatic number, i.e. each containing a fixed number of cycles (besides the trees attached), for a given graph. The enumeration of spanning trees for a labelled graph is usually computed by means of the adjaeeey matrix of the given graph. On the other hand, in contrast to the adjacency matrix method (or essentially the "incidence matrix" method), we approached the problem some years ago by a dual notion and derived a computational expression using the concept of cycles in defining the required matrix entries [1]. The duality is, of course, referred to that between the vertices and the cycles; the matrix entries are indexed by the vertices of the given graph in the adjacency matrix method, while they are indexed by the assigned cycles in the latter approach. It was already apparent to us, at that time, that the computational effectiveness of these two approaches depends critically on the nature of the given graph. Some examples were given in that paper [1] to point out that the adjacency matrix method is clearly not as effective as the "cycle matrix" method if the given graph involves many vertices but very few cycles (and vice versa). However, the formal expressions derived by either approach are of equal simplicity and elegance. In the present investigation, we rely on the concept of cycles. However, a direct application of cycle matrices [1] does not appear to be very effective. As it turns out, the problems can be handled efficiently by introducing the so-called cycle-adjacency matrix for a given connected graph after labelling the cycles considered. In carrying out the dual notion to the usual adjacency-matrix, it is necessary to impose the requirement that each edge of the graph can belong at most to two independent cycles. Using this matrix, together with some further auxiliary notions, we derive the explicit expressions for the enumeration of the connected subgraphs (of a planar graph) each containing one and two cycles. These explicit expressions suggest immediately the general expression for n cycles. Though it is natural to try to prove it by a mathematical induction on n, yet the involvement of determinants makes it computationally very complicated. We resolve this by introducing the i-th "annihilation operator" which deletes the i-th column and the i-th row in the cycle-adjacency matrix. Together with a formal procedure, the operator method provides a straight-