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# Lecture05 - CSE 421 Algorithms Richard Anderson Lecture 5...

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CSE 421 Algorithms Richard Anderson Lecture 5 Graph Theory

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Bipartite A graph is bipartite if its vertices can be partitioned into two sets V 1 and V 2 such that all edges go between V 1 and V 2 A graph is bipartite if it can be two colored
Theorem: A graph is bipartite if and only if it has no odd cycles If a graph has an odd cycle it is not bipartite If a graph does not have an odd cycle, it is bipartite

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Lemma 1 If a graph contains an odd cycle, it is not bipartite
Lemma 2 If a BFS tree has an intra-level edge , then the graph has an odd length cycle Intra-level edge: both end points are in the same level

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Lemma 3 If a graph has no odd length cycles, then it is bipartite No odd length cycles implies no intra-level edges No intra-level edges implies two colorability
Connected Components Undirected Graphs

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Computing Connected Components in O(n+m) time A search algorithm from a vertex v can find all vertices in v’s component While there is an unvisited vertex v, search from v to find a new component
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