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Unformatted text preview: MATH 682 Notes Combinatorics and Graph Theory II 1 Flows A logical followup (or predecessor) to Mengers theorem, and its associated discussion of multiple simultaneous routes, is the concept of flow in a graph. If we visualize a graph as a system of pipelines between a source and a sink , then a significant question becomes how to route whatever it is were transferring to make optimal use of our network. To properly address flow, well actually need to make a few changes to the concept of a graph to generalize it better. Definition 1. A system D = ( V,E ) where V is a set of vertices and E is a set of ordered pairs of elements of V is called a directed graph or digraph . The edge ( u,v ) is said to point from u to v . Many of the concepts defined for graphs have analogues in directed graphs. For instance, a directed walk is a sequence of edges ( v ,v 1 ) , ( v 1 ,v 2 ) ,..., ( v k- 1 ,v k ), frequently denoted v v 1 v k . There are likewise concepts of directed trails and paths, and of connectivity by means of directed paths, called strong connectivity . Digraphs also possess twovarieties of the concept of vertex degree: the indegree d- ( v ) counting incoming edges, and the outdegree d + ( v ) counting outgoing edges. We will develop other, novel ideas associated with directed graphs in greater detail after discussing flows. Another concept necessary when discussing flow is weight , which is an assignment of values to graph features, usually edges, but occasionally vertices: Definition 2. An edge-weight for a (di)graph G is a function f : E ( G ) R , and a vertex-weight is a function f : V ( G ) R . A (di)graph together with a weight of either type is called a weighted (di)graph . Weights are usually, although not necessarily, restricted to non-negative values. Weight can rep- resent a number of different edge properties in practice: they can be utilization cost, length of connection (in which case the concept of distance is frequently redefined), quality of connection, or possible throughput. It is the last of these properties that we shall address with regard to flow. The edges of a graph could be thought of as pipelines for material, while the vertices are stations for management and routing. With this model in mind, a question with clear real-world applications is: what is the largest quantity of material we can transfer this way, and how do we do so? We shall formally pose this question by indicating which transmission plans are legal in the sense of actually representing a viable flow. For semantic simplicity, below we shall refer to the edge- weights in a weighted digraph as capacities, and denote the capacity of an edge e by c ( e ); for simplicity, we shall also use c ( u,v ) to denote capacity on the edge ( u,v ) instead of the more formal...
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