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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]

Given an edge-weighted undirected graph G = (V , E , w) and a subset T ⊆ V of p terminals, a k-Steiner forest spanning all the terminals in T includes k branches, where every branch is a Steiner tree. The diameter of a k -Steiner forest is referred to as the maximum distance between two terminals of a branch. This paper studies the minimum diameter k-Steiner forest problem (MD k SFP) and establishes the relationship between MD k SFP and the absolute k-Steiner center problem (A k SCP). We first obtain a 2-factor dual approximation algorithm for A k SCP, and then achieve a 2-approximation algorithm for MD k SFP based on the 2-approximation to A k SCP. Furthermore, we develop an improved 2 ρ -approximation algorithm for MD k SFP, where ρ < 1 in general, by perturbing the sites of facilities and re-clustering the terminals. [ABSTRACT FROM AUTHOR]