Here are some counterexamples in my answers.
Rooted Tree Isomorphism Algorithm
Local search to find minimum degree spanning tree
Steiner tree problem in graphs of diameter 3
Does every infinite context free language contain an infinite regular subset?
Is a set of acyclic |V| - 1 light edges always a Minimum Spanning Tree?
Gerrymandering Problem: Variant on Set Partitioning
Finding the longest path in an undirected node-weighted tree
All superlinear runtime algorithms are asymptotically equivalent to convex function?
Does an optimal path imply the heuristic is admissible?
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