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I deg i is the number of arcs incident to i for

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i, deg ( i ), is the number of arcs incident to i . For directed graphs, in-degree and out-degree, indeg ( i ) and outdeg ( i ), are the number of arcs pointing into and out of i , respectively . In a dense network, deg ( i ) ~ O(| N |) for most i , and hence | E |=O(| N | 2 ). In a sparse network, deg ( i ) ~ O(1) for most i , and hence | E |=O(| N |). PATH : A path is a sequence of nodes { i 1 , i 2 , i 3 ,…, i k } and relevant arcs such that ( i j , i j+ 1 ) A and no nodes are repeated. CYCLE : A cycle is a path + the edge ( i k , i 1 ). CONNECTED NETWORK : G is a connected network if a path exists between any pair of nodes. TREE : A tree is a graph without cycles. SPANNING TREE : A spanning tree is a connected subgraph that includes all nodes and is a tree. Example: (thick links form a spanning tree) 1 2 3 4

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Logistics Systems Analysis Basics on Graphs and Networks MINIMUM SPANNING TREE (MST) Among all the spanning trees of a weighted (cost) and connected graph G = ( N , E ), the one (or possibly more) with the least total weight is called a minimum spanning tree (MST). Two algorithms: • Kruskal's Algorithm 1. Let edge set E 0 = . Find the cheapest edge ( i,j ) in the graph (if there is more than one, pick one at random) and include it into E 0 2. Find the cheapest edge in E \ E 0 that doesn't close a cycle with those already chosen in E 0 . Include it into E 0 . 3. Repeat Step 2 until you reach out to every vertex of the graph (or | E 0 | = n –1). • Prim's Algorithm 1. Pick any vertex as a starting vertex set. (Call it S ). Let E 0 = . 2. Find the nearest neighboring vertices k S and p N \ S , such that c ( p,k ) = min{ c ( i,j ), i S , j N \ S }. Let S S { p }, E 0 E 0 {( p,k )}. 3. Repeat Step 2 until all vertices are in S , e.g. | S | = n . EULER TOUR An Euler tour in an undirected graph G is a cycle using every edge exactly once. An Euler tour in a directed graph G is a directed cycle which includes every arc exactly once. A connected undirected graph has an Euler tour if and only if the degree of every vertex is even. A weakly connected directed graph has an Euler tour if and only if for any vertex v , indeg(v) = outdeg(v). Euler’s famous “test problem:” can one find out whether or not it is possible to cross each bridge exactly once?" Complexity O( n 2 ) Complexity O( n 2 log(n)) Logistics Systems Analysis Basics on Graphs and Networks MATCHING A matching in a graph G =( V , E ) is a subset M of the edges E such that no two edges in M share a common end node. A perfect matching M in G is a matching such that each node of G is incident to an edge in M . If the edges of the graph have an associated weight, then a maximum (minimum) weighted perfect matching is a perfect matching such that the sum of the weights of the edges in the matching is maximum (minimum). There is a complicated but polynomial-time algorithm (e.g., O( n 2 m ) ~ blossom algorithm by Edmonds 1965; O( n 3 ) ~ Gabow, 1975; Lawler, 1976) for finding a minimum weight perfect matching.
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