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Unformatted text preview: MATH 682 Notes Combinatorics and Graph Theory II 1 Local and Global 2-connectedness 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 2-connected, 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 end-to-end: 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 2-connectivity 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 walk-construction 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 end-to-end: 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 2-connected, 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 2-connected graphs joined along a skeleton, which is in fact a tree. We can, if we like, use tihs to build a block-structural version of the structure theorem for 1-connected 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 2-connected or a single edge....
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