Davide Cherubini - PhD Thesis - UniCA Eprints

6.2 Graphs and Network Flowso a cost denoting the “price” to be paid by a single commodity unit totraverse the arc;o an upper and a lower capacity constraint denoting the maximum andthe minimum amount of commodity that can be carried by the arc.A graph can be undirected if its arcs have no direction (edges or lines), or itcan be a directed graph if each of its arcs is directed from a node x to a node y(arcs, directed edges, or arrows). In this case y is called the head and x is calledthe tail of the edge; y is said to be a direct successor of x, and x is said to be adirect predecessor of y.It is useful to define the concept of Backward Star and Forward Star. Theformer represents the set of edges entering in a node, while the latter is the setof edges outgoing from a node.The incidence matrix E for directed graphs is a [n × m] matrix, where n and mare the number of nodes and arcs respectively, such that:are:⎧⎨ −1 if the arc a j leaves the node n iE ij = 1 if the arc a j enters the node n i⎩0 otherwiseIt is possible to define three different types of problem over a graph, which1. the minimum cost flow problem2. the maximum flow problem3. the shortest path problem6.2.1 The Minimum Cost Flow ProblemLet G = (N, A) be a directed graph and x ∈ R m be a vector, where m is thenumber of edges. x is said to be a flow for G if it verifies the flow conservationequation constraint:∑x ji −∑x ij = b i i ∈ N (6.4)(j,i)∈BS(i) (i,j)∈F S(i)45