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Unformatted text preview: MATH 682 Notes Combinatorics and Graph Theory II 1 Local and Global 2connectedness 1.1 Block structure, concluded Previously we saw that the block structure of a connected graph was connected; today we shall see that it is in fact a tree. Proposition 1. Given a connected graph G with block diagram B ( G ) , B ( G ) is acyclic. Proof. Suppose B ( G ) contains the minimal cycle B 1 ∼ v 1 ∼ B 2 ∼ v 2 ∼ B 3 ∼ ··· ∼ B n ∼ v n ∼ B 1 . We shall show that the subgraph B = B 1 ∪ B 2 ∪ B 3 ∪ ··· ∪ B n of G is 2connected, violating the maximality condition of the blocks B i . Note that B is connected, since if u ∈ B i and v ∈ B j , and without loss of generality i ≤ j , we may construct a walk from u to v via one of the two following scenarios. If i = j , then we simply construct a path inside of B i between the two. If i < j , then we build our walk by placing the following paths endtoend: the path from u to v i guaranteed by connectivity of B i , the path from v i to v i +1 guaranteed by connectivity of B i +1 , and so forth up to the path from v j 2 to v j 1 guaranteed by connectivity of B j 1 , and then a path from v j 1 to v guaranteed by connectivity of B j . Now, we shall see that such a walk will remain even if an arbitrary vertex w other than u or v themselves is removed from B . There are several possible vertices w , msot of which have no effect on the construction above: if w ∈ B k but w 6 = v k 1 ,v k , then by 2connectivity of individual blocks, B k w is still connected, and all of the named vertices mentioned in the procedure above are still under consideration. Likewise, removal of any v k with k < i or k ≥ j will have no effect on the walkconstruction procedure above. Our only concern, then, is removal of a vertex v k with i ≤ k < j . In this case, we can construct a walk with the opposite routing arond the cycle, by placing the following paths endtoend: the path from u to v i 1 guaranteed by connectivity of B i , the path from v i 1 to v i 2 guaranteed by connectivity of B i 1 , and so forth through the path from v 2 to v 1 guaranteed by connectivity of B 2 , and then onto the path from v 1 to v k guaranteed by connectivity of B 1 , continuing to descend to the path from v j +1 to v j guaranteed by connectivity of B j +1 , and then a path from v j to v guaranteed by connectivity of B j . Thus, B is 2connected, so the ostensible blocks within it are not blocks, contradicting the possibility of a cycle of blocks. We thus know that a connected graph consists of a number of 2connected graphs joined along a “skeleton”, which is in fact a tree. We can, if we like, use tihs to build a blockstructural version of the structure theorem for 1connected graphs: Proposition 2. For any connected graph G , there is a sequence of connected subgraphs G 1 ⊂ G 2 ⊂ ··· ⊂ G n = G such that: • G 1 is 2connected or a single edge....
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This note was uploaded on 01/12/2012 for the course MATH 682 taught by Professor Wildstrom during the Spring '09 term at University of Louisville.
 Spring '09
 WILDSTROM
 Combinatorics, Graph Theory

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