2.3.2 Let T be a tree with at least k edges, k ? 2. How many connected components are there in the subgraph of T obtained by deleting k edges of T? 2.3.3 Let G be a connected graph which is not a tree and let C be a cycle in G. Prove that the complement of any spanning tree of G contains at least one edge of C.
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2.3.2 Let T be a tree with at least k edges. How many connected components are there in the subgraph of T obtained by deleting k edges of T? 2.3.3 Let G be a connected graph which is not a tree, and let C be a cycle in G. Prove that the complement of any spanning tree of G contains at least one edge of C.
Sri K.
Prove that every connected graph G has a spanning tree as follows: a. Start with the graph G and work down by deleting edges until you have a spanning tree. b. Start with the vertices of the graph G and work up by adding edges from G until you have a spanning tree.
Prove that for each simple graph G the following are equivalent. a) Every induced subgraph of G has a vertex of degree at most 1. b) The intersection of any two intersecting paths in G is a path. c) The number of components of G is the number of vertices minus the number of edges. d) G is a forest (i.e. a simple graph with no cycles).
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