Classification of Weighted Graphs up to Blowing-up and Blowing-down

We classify weighted forests up to the blowing-up and blowing-down operations which are relevant for the study of algebraic surfaces. The word “graph” in this text means a finite undirected graph such that no edge connects a vertex to itself and at most one edge joins any given pair of vertices. A weighted graph is a graph in which each vertex is assigned an integer (called its weight). Two operations are performed on weighted graphs: The blowing-up and its inverse, the blowing-down. Two weighted graphs are said to be equivalent if one can be obtained from the other by means of a finite sequence of blowings-up and blowings-down (see 1.4–1.7). These weighted graphs and operations are well known to geometers who study algebraic surfaces. Many problems in the geometry of surfaces can be formulated in graph-theoretic terms and solving these sometimes requires elaborate graph-theoretic considerations. This gives rise to a variety of questions about weighted graphs, all in connection with the equivalence relation generated by blowing-up and blowing-down. The present paper proposes a classification of weighted forests up to equivalence. In particular, Theorem 8.34 defines an invariant Q̄(G) for any pseudo-minimal (3.8) weighted forest G, and asserts that Q̄(G) = Q̄(G) if and only if G is equivalent to G. Since Q̄(G) can actually be computed, this yields an algorithm for deciding whether two weighted forests are equivalent (see Problem 5, at the end of section 8). Apparently this decision problem was previously open, even in the special case of “linear chains”, i.e., weighted graphs of the form: r r . . . r x1 x2 xq (xi ∈ Z). Note that, in the case of linear chains, 8.34 simplifies to 5.4. We also contribute to the problem of listing all minimal elements in a given equivalence class of weighted forests. Section 9 reduces that problem to the case of linear chains. This special case is given a recursive solution in Section 7 and, in some simple cases, an explicit solution. Incidentally, the cases that we are able to describe explicitely are precisely those which arise from the study of algebraic surfaces. Section 3 is concerned with topological properties of graphs, but topology is never mentioned. For instance, 3.6 has the following consequence: Let G,G be weighted trees which are minimal and equivalent. If G is not a linear chain then the two trees are homeomorphic.