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Unformatted text preview: Algorithms Lecture 11: Basic Graph Properties Obie looked at the seein’ eye dog. Then at the twenty-seven 8 by 10 color glossy pictures with the circles and arrows and a paragraph on the back of each one...and then he looked at the seein’ eye dog. And then at the twenty-seven 8 by 10 color glossy pictures with the circles and arrows and a paragraph on the back of each one and began to cry. Because Obie came to the realization that it was a typical case of American blind justice, and there wasn’t nothin’ he could do about it, and the judge wasn’t gonna look at the twenty- seven 8 by 10 color glossy pictures with the circles and arrows and a paragraph on the back of each one explainin’ what each one was, to be used as evidence against us. And we was fined fifty dollars and had to pick up the garbage. In the snow. But that’s not what I’m here to tell you about. — Arlo Guthrie, “Alice’s Restaurant” (1966) I study my Bible as I gather apples. First I shake the whole tree, that the ripest might fall. Then I climb the tree and shake each limb, and then each branch and then each twig, and then I look under each leaf. — Martin Luther 11 Basic Graph Properties 11.1 Definitions A graph G is a pair of sets ( V , E ) . V is a set of arbitrary objects that we call vertices 1 or nodes . E is a set of vertex pairs, which we call edges or occasionally arcs . In an undirected graph, the edges are unordered pairs, or just sets of two vertices. In a directed graph, the edges are ordered pairs of vertices. We will only be concerned with simple graphs, where there is no edge from a vertex to itself and there is at most one edge from any vertex to any other. Following standard (but admittedly confusing) practice, I’ll also use V to denote the number of vertices in a graph, and E to denote the number of edges. Thus, in an undirected graph, we have ≤ E ≤ V 2 , and in a directed graph, 0 ≤ E ≤ V ( V- 1 ) . We usually visualize graphs by looking at an embedding . An embedding of a graph maps each vertex to a point in the plane and each edge to a curve or straight line segment between the two vertices. A graph is planar if it has an embedding where no two edges cross. The same graph can have many different embeddings, so it is important not to confuse a particular embedding with the graph itself. In particular, planar graphs can have non-planar embeddings! a b e d f g h i c a b e d f g h i c A non-planar embedding of a planar graph with nine vertices, thirteen edges, and two connected components, and a planar embedding of the same graph. 1 The singular of ‘vertices’ is vertex . The singular of ‘matrices’ is matrix . Unless you’re speaking Italian, there is no such thing as a vertice, a matrice, an indice, an appendice, a helice, an apice, a vortice, a radice, a simplice, a codice, a directrice, a dominatrice, a Unice, a Kleenice, an Asterice, an Obelice, a Dogmatice, a Getafice, a Cacofonice, a Vitalstatistice, a Geriatrice, or Jimi Hendrice! Youor Jimi Hendrice!...
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- Spring '09
- Graph Theory, Basic Graph Properties