Decomposition of graphs on surfaces and a homotopic circulation theorem

Abstract

We prove the following theorem. Let G be an eulerian graph embedded (without crossings) on a compact orientable surface S. Then the edges of G can be decomposed into cycles C 1,…, C t in such a way that for each closed curve D on S mincr(G,D)= ∑ i=1 l mincr(C i,D) . Here mincr ( G, D) denotes the minimum number of crossings of G and D ̃ , among all closed cures D ̃ homotopic to D (such that D ̃ does not intersect vertices of G). Similarly, mincr ( C, D) denotes the minimum number of crossings of C ̃ and D ̃ , among all closed curves C ̃ and D ̃ homotopic to C and D, respectively. As a corollary we derive the following “homotopic circulation theorem.” Let G = ( V, E) be a graph embedded on a compact orientable surface S, let c: E → Q + be a “capacity” function, let C 1,…, C k be cycles in G, and let d 1,…, d k ϵ Q + be “demands.” Then there exist circulations x 1,…, x k in G such that each x i decomposes fractionally into d i cycles homotopic to C i ( i = 1,…, k) and such that the total flow through any edge does not exceed its capacity, if and only if for each closed curve D on S which does not intersect vertices of G we have that the sum of the capacities of the edges intersected by D (counting multiplicities) is not smaller than Σ i = 1 k d i ·mincr( C i , D). This applies to a problem posed by K: Mehlhorn in relation to the automatic design of integrated circuits.