Applied_Combinatorics_chapter_1

Applied_Combinatorics_chapter_1 - C HAPTERl ELEMENTS OF...

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1.1 CHAPTERl ELEMENTS OF GRAPH THEORY GRAPH MODELS The first four chapters deal with graphs and their applications. A graph G = (V, E) consists of a finite set V of vertices and a set E of edges joining different pairs of distinct vertices. * Figure 1.1 a shows a depiction of a graph with V = {a, b, c, d} and E = ((a, b), (a, c), (a, d), (b, d), (c, d)}. We represent vertices with points and edges with lines joining the prescribed pairs of vertices. This definition of a graph does not allow two edges to join the same two vertices. Also an edge cannot "loop" so that both ends terminate at the same vertex-an edge's end vertices must be distinct. The two ends of an undirected edge can be written in either order, (b, c) or (c, b). We say that vertices a and b are adjacent when there is an edge (a, b). Sometimes the edges are ordered pairs of vertices, called directed edges. In a directed graph, all edges are directed. See the directed graph in Figure 1.1 b. We write (b~ c) to denote a directed edge from b to c. In a directed graph, we allow one edge in each direction between a pair of vertices. See edges (a ~ c) and (c~ a) in Figure l.lb. The combinatorial reasoning required in graph theory, and later in the enumera- tion part of this book, involves different types of analysis than used in calculus and high school mathematics. There are few general rules or formulas for solving these problems. Instead, each question usually requires its own particular analysis. This analysis sometimes calls for clever model building or creative thinking but more of- ten consists of breaking the problem into many cases (and subcases) that are easy enough to solve with simple logic or basic counting rules. A related line of reason- ing is to solve a special case of the given problem and then to find ways to extend that reasoning to all the other cases that may arise. In graph theory, combinatorial arguments are made a little easier by the use of pictures of the graphs. For example, a case-by-case argument is much easier to construct when one can draw a graphical depiction of each case. Graphs have proven to be an extremely useful tool for analyzing situations in- volving a set of elements in which various pairs of elements are related by some property. The most obvious examples of graphs are sets with physical links, such 'What this book calls a graph is referred to in many graph theory books as a simple graph. In general, graph theory terminology varies a little from book to book. ---.
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1:1'1 1 ,I '1,:1 1 1;1 I I 'I ,' b "LSJ, "\S7\ d (a) (b) Figure 1.1 as electrical networks, where electrical components (transistors) are the vertices and connecting wires are the edges; or telephone communication systems, where tele- phones and switching centers are the vertices and telephone lines are the edges. Road maps, oil pipelines, and subway systems are other examples.
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Applied_Combinatorics_chapter_1 - C HAPTERl ELEMENTS OF...

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