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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 [Bilò, ISAAC 2018]. • The paper presents a linear time algorithm for computing the diameter of unicycle graphs. • The paper gives a new algorithm which finds a new edge to add to a tree so that the diameter of the new graph is minimized. • The algorithms rely on certain domination relations among edges of the graphs. [ABSTRACT FROM AUTHOR]

In the 2-Club Cluster Vertex Deletion (resp., 2-Club Cluster Edge Deletion) problem the input is a graph G and an integer k , and the goal is to decide whether there is a set of at most k vertices (resp., edges) whose removal from G results in a graph in which the diameter of every connected component is at most 2. In this paper we give algorithms for 2-Club Cluster Vertex Deletion and 2-Club Cluster Edge Deletion whose running times are O ⁎ (3.104 k) and O ⁎ (2.562 k) , respectively. Our algorithms were obtained using automated generation of branching rules. Our results improve the previous O ⁎ (3.303 k) -time algorithm for 2-Club Cluster Vertex Deletion [Liu et al., FAW-AAIM 2012] and the O ⁎ (2.695 k) -time algorithm for 2-Club Cluster Edge Deletion [Abu-Khzam et al., TCS 2023]. [ABSTRACT FROM AUTHOR]