We now consider undirected graphs. Recall that such a graph is • connected iff for all...
We now consider undirected graphs. Recall that such a graph is
• connected iff for all pairs of nodes u, w, there is a path of edges between u and w;
• acyclic iff for all pairs of nodes u, w, whenever there is an edge between u and w then there is no path
Given an acyclic undirected graph G with n nodes (where n ≥ 1)
and a edges, your task is to prove that a ≤ n − 1, and that
equality holds (a = n − 1) if and only if G is connected.
Hint: a possible approach is to do induction in a; for the case
with a > 0, consider an edge {x, y} and partition G into 3
parts: the nodes connected to x (but not through that edge), those
connected to y (but not through that edge), and the remaining
nodes. Then apply the induction hypothesis to each part.
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We now consider undirected graphs. Recall that such a graph
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