17-graphs - Algorithms Lecture 17 Basic Graph Properties...

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Unformatted text preview: Algorithms Lecture 17: Basic Graph Properties [ Sp’10 ] 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 17 Basic Graph Properties 17.1 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; I will usually write uv instead of { u . v } to denote the undirected edge between u and v . In a directed graph, the edges are ordered pairs of vertices; I will usually write u v instead of ( u . v ) to denote the directed edge from u to v . We will usually be concerned only 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 ) . If ( u , v ) is an edge in an undirected graph, then u is a neighbor or v and vice versa. The degree of a node is the number of neighbors. In directed graphs, we have two kinds of neighbors. If u v is a directed edge, then u is a predecessor of v and v is a successor of u . The in-degree of a node is the number of predecessors, which is the same as the number of edges going into the node. The out-degree is the number of successors, or the number of edges going out of the node. A graph G = ( V , E ) is a subgraph of G = ( V , E ) if V ⊆ V and E ⊆ E ....
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17-graphs - Algorithms Lecture 17 Basic Graph Properties...

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