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graphs - CS 70 Fall 2006 Graphs Discrete Mathematics for CS...

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CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 16 Graphs Formulating a simple, precise specification of a computational problem is often a prerequisite to writing a computer program for solving the problem. Many computational problems are best stated in terms of graphs: a directed graph G ( V , E ) consists of a finite set of vertices V and a set of (directed) edges or arcs E . An edge is an ordered pair of vertices ( v , w ) and is usually indicated by drawing a line between v and w , with an arrow pointing towards w . Undirected graphs may be regarded as special kinds of directed graphs, such that ( u , v ) E ( v , u ) E . Thus, since the directions of the edges are unimportant, an undirected graph G ( V , E ) consists of a finite set of vertices V , and a set of edges E , each of which is an unordered pair of vertices { u , v } . As we have defined them, graphs are allowed to have self-loops; i.e. edges of the form ( u , u ) that go from a vertex to itself. Sometimes it is more convenient to disallow such self-loops. Graphs are useful for modeling a diverse number of situations. For example, the vertices of a graph might represent cities, and edges might represent highways that connect them. In this case, the edges would be undirected: In the above picture, V = { SF , LA , NY ,... } , and E = {{ SF , LA } , { SF , NY } ,... } . Alternatively, an edge might represent a flight from one city to another, and now edges would be directed (a certain airline might have a non-stop flight from SFO to LAX, but no non-stop flight back from LAX to SFO). Graphs can also be used to represent more abstract relationships. For example, the vertices of a graph might represent tasks, and the edges might represent precedence constraints: a directed edge from u to v says that task u must be completed before v can be started. An important problem in this context is scheduling: in what order should the tasks be scheduled so that all the precedence constraints are satisfied.
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