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].

中文翻译：

---

独轮车图和直径最优增广树的直径算法

我们考虑计算独轮图（即具有唯一环的图）的直径的问题。我们提出了一个 O(n) 时间算法来解决这个问题，其中 n 是图的顶点数。这改进了之前最好的 O(n \log n) 时间解决方案 [Oh and Ahn, ISAAC 2016]。使用该算法作为子程序，我们解决了向树添加快捷方式的问题，从而使新图（即独轮图）的直径最小；我们的算法需要 O(n^2 \log n) 时间和 O(n) 空间。以前最好的算法在 O(n^2 \log^3 n) 时间和 O(n) 空间中解决问题 [Oh and Ahn, ISAAC 2016]，或在 O(n^2) 时间和 O(n^2 ) 空间 [Bil\`o, ISAAC 2018]。