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Unformatted text preview: Graph isomorphism/connectivity Margaret M. Fleck 16 April 2010 This lecture finishes our coverage of basic graph concepts: isomorphism, paths, and connectivity. It shows the basic ideas without covering every possible permutation of these ideas e.g. for different types of graphs. 1 Hypercubes Recall from last class that the ncube or a hypercube Q n is the graph of the corners and edges of an ndimensional cube. It is defined recursively as follows (for any n ∈ N ): 1. Q is a single vertex with no edges 2. Q n consists of two copies of Q n 1 with edges joining corresponding vertices. The hypercube defines a binary coordinate system. To build it, we label nodes with binary numbers, where each binary digit corresponds to the value of one coordinate. 011 010 001 000 111 110 101 100 1 Q n has 2 n nodes. To compute the number of edges, we set up a recurrence: 1. E (0) = 0 2. E ( n ) = 2 E ( n 1) + 2 n 1 The 2 n 1 term is the number of nodes in each copy of Q n 1 , i.e. the number of edges required to join corresponding nodes. 2 Bipartite graphs The last special type of graph is a bipartite graph. A graph G = ( V, E ) is bipartite if we can split V into two nonoverlapping subsets V 1 and V 2 such that every edge in G connects an element of V 1 with an element of V 2 . That is, no edge connects two nodes from the same part of the division. Bipartite graphs often appear in matching problems, where the two subsets represent different types of objects, e.g. matching a group of women with a group ofdifferent types of objects, e....
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This note was uploaded on 12/24/2010 for the course CS CS 173 taught by Professor Fleck during the Spring '10 term at University of Illinois, Urbana Champaign.
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