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 is • connected iff for all...
P9.6.3 Prove that a connected undirected graph G is bipartite if and only if there are no edges between nodes at the same level in any BFS tree for G. (An undirected graph is defined to be bipartite if its nodes can be divided into two sets X and Y such that all edges have one endpoint in X and the other in Y.) P9.6.3 Prove that a connected undirected graph G is bipartite if and only if there are...
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