We consider the problem of computing the diameter of a unicycle graph (i.e., a graph with a unique cycle). We present an O(n) time algorithm for the problem, where n is the number of vertices of the graph. This improves the previous best O(n log n) time solution [Oh and Ahn, ISAAC 2016]. Using this algorithm as a subroutine, we solve the problem of adding a shortcut to a tree so that the diameter of the new graph (which is a unicycle graph) is minimized; our algorithm takes O(n^2 log n) time and O(n) space. The previous best algorithms solve the problem in O(n^2 log^3 n) time and O(n) space [Oh and Ahn, ISAAC 2016], or in O(n^2) time and O(n^2) space [Bil`o, ISAAC 2018].
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