Combinatorial optimization: networks and matroids by Lawler E.L.

By Lawler E.L.

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Some of the representations of graphs that are appropriate for computers are an arc list, an incidence matrix, and an adjacency matrix. 1 Two drawings of the same graph Graphs and Digraphs 21 An arc list simply contains an entry for each arc (i, j). 4) (3, 5) (4,5). Arc lists may be sorted, ordered, and manipulated in various ways within the computer. An arc (i, j) is said to be incident to each of the nodes i and j, and conversely. Each row of the node-arc incidence matrix is identified with a node and each column with an arc.

This is an optimal feasible basis, since no increase in the values of the nonbasic variables can further reduce the value of z. Each feasible basis uniquely determines a value of z. At each pivot step. z is decreased by a finite amount (provided X, > 0 for that pivot step). Thus no feasible basis can be repeakd. Since there are a finite number of possible bases, the procedure must terminate with an optimal solution after a finite number of pivot steps. A number of technical questions still remain.

An (s, t) path is open if s # t and closed if s = t. A cq’cle is an (s, s) path containing at least one arc, in w:hich no node except s is repeated. In an ordinary graph (as opposed to a multigraph or ,a graph with loops), a cycle must contain at least three arcs. A graph which contains no cycles is acyclic. Two nodes i and j are said to be connected if there exists an (i, j) path. A graph G is said to be connected if all pairs of nodes are connected. , it is not a subgraph of any other connected subgraph of G.

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