CS300-10_Graph_and_Digraph

# CS300-10_Graph_and_Digraph - 5 Biconnected Components of A...

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5. Biconnected Components of A Graph If one city’s airport is closed by bad weather, can you still fly between any other pair of cities? If one computer in a network goes down, can a message be sent between any other pair of computers? If any one vertex (and edges incident with it) is removed from a connected graph, is the remaining subgraph still connected?

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G I E C F B A D H J F A D H J E F E C B G I B (a) graph. (b) Its biconnected components Def’n : Let G= ( V, E ) be a connected, undirected graph. A vertex a V is said to be an articulation point if there exist vertices v and w such that (1) v, w and a are distinct (2) Every path between v and w must contain a . Alternatively, a is an articulation point of G if removing a splits G into two or more parts. subgraphs
Def’n: A graph G = ( V, E ) is said to be biconnected if and only if it has no articulation points. G 1 G 2 G 3 Which of the above are biconnected? Def’n: Let G' = (V', E') be a biconnected subgraph of a graph G = (V, E). G' is said to be a biconnected component of G if G' is maximal i.e., not contained in any other biconnected subgraph of G . Examples are shown in the previous page !!!

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G = ( V, E ) Two edges e 1 and e 2 in E are said to be related, i.e., ( e 1 , e 2 ) R if e 1 = e 2 or if there is a cycle containing both e 1 and e 2 . ( e, e ) R 2200 e E ( e 1 , e 2 ) R and ( e 2 , e 3 ) R ( e 1 , e 3 ) R ( e 1 , e 2 ) R ( e 2 , e 1 ) R why ? Observation e 1 e 1 e 3 Why transitivity?
An equivalence relation on E partitions E into equivalence classes !!! Each subgraph consisting of the edges in an equivalence class and the incident vertices form a biconnected component !!! Can you prove it ? Homework.

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G i Lemma : For 1 i k , Let G i =( V i , E i ) be the biconnected components of a connected undirected graph G = ( V , E ). Then (1) G i is biconnected for each 1 i k , (2) For all i j , V i V j contains at most one vertex. (3) a is an articulation point of G if and only if a V i V j for some i j [ proof ] (1) Trivial why ? (2) suppose that two distinct vertices v and w are in V i V j , i j v w C 1 G j C 2 Why? (3) x a y ( ) ( ) x a y v w Homework.
A B C D E A B C D E F A B C D E A B C D E F A is not an articulation point A is an articulation point Can you now characterize an articulation point when A is the root ?

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A B C D E F G H I A B C D E F G H I Can you characterize D ?
An articulation point v in a depth-first search tree. Every path from root to w passes through v . v w root

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Theorem : Let G = ( V, E ) be a connected, undirected graph and let S = ( V, T ) be a depth first spanning tree for G . Vertex a is an articulation point of G if and only if one of the following is true: (1) a is the root and a has two or more sons. (2) a is not the root and for some son s of a , there is no back edge between any descendant of s ( including s itself ) and a proper ancestor of a .
[proof] i) The root is an articulation point if and only if it has two or more sons.

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CS300-10_Graph_and_Digraph - 5 Biconnected Components of A...

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