Flowing Graphs

Abstract

This chapter explains how to assign weights to the nodes in an edge-weighted graph or to the edges in a node-weighted graph. In a node-weighted graph, each edge is the flowing edge of one of its extremities, whose weight may be assigned to the edge. In an edge-weighted graph, each node is the origin of one or several flowing edges, the adjacent edges with minimal weight. The chapter models node- and edge-weighted graphs as a network of tanks interconnected by pipes. The level of water in the tanks represents the node weights and the altitude of the pipes represents the edge weights. The chapter establishes an equivalence between node- and edge-weighted graphs. It associates each node (respectively edge) weighted graph with an edge (respectively node) weighted graph such that the regional minima and catchment zones of both graphs are identical.