Towers and Cycle Covers for Max-Balanced Graphs

Let G = (V,A,g) be a strongly connected weighted graph. We say that G is max-balanced if for every cut W, the maximum weight over arcs leaving W equals the maximum weight over arcs entering W. A subgraph H of G is max-sufficient if (or every cut W, the maximum weight over a.rcs of G leaving W is attained at some arc of H. A tower T = (CI,C., ... ,C,) is a sequence of arc-sets of G where C'+ I is a cycle all of whose weights are maximal in the graph formed by contracting the sets CI , C., ... , C; to a point. We show that G is maxbalanced if and only if G contains a tower. A cycle caver for G is a collection of cycles V = {D. I a E A} such that arc a is the minimum weight arc of Do . We use the tower construction to show that the existence of a cycle cover chara.cterizes max-balanced graphs. We show that the graph H of a tower is max-sufficient , thereby showing that a max-balanced graph contains a max-sufficient subgraph with at most 2(1V1 1) arcs. Further , we use the tower construction to show that H has a cycle cover with at most IVI cycles.