Given an undirected graph G = (V, E) and an integer k, the s-Club asks if Gcontains a vertex subset S of at least k vertices such that G[S] has diameter at most s. Recently, Vertex r-Triangle s-Club, and Edge r-Triangle s-Club that generalize the notion of s-Club have been studied by Garvardt et al. [TOCS-2023, IWOCA-2022] from the perspective of parameterized complexity. Given a graph G and an integer k, the Vertex r-Triangle s-Club asks if there is an s-Club S with at least k vertices such that every vertex u \in S is part of at least r triangles in G[S]. In this paper, we initiate a systematic study of Vertex r-Triangle s-Club for every integer r >= 1 from the perspective of structural parameters of the input graph. In particular, we provide FPT algorithms for Vertex r-Triangle 2-Club when parameterized by the treewidth (tw) of the input graph, and an XP algorithm when parameterized by the h-index of the input graph. Additionally, when parameterized by the feedback edge number (fes) of the input graph. We provide a kernel of O(fes) edges for Vertex r-Triangle s-Club.

翻译：给定一个无向图G=(V,E)和一个整数k，s-团问题询问G是否包含一个顶点子集S（|S|≥k），使得诱导子图G[S]的直径不超过s。近年来，Garvardt等人[TOCS-2023, IWOCA-2022]从参数化复杂性的角度研究了推广s-团概念的顶点r-三角形s-团和边r-三角形s-团问题。给定图G和整数k，顶点r-三角形s-团问题询问是否存在一个包含至少k个顶点的s-团S，使得每个顶点u∈S在G[S]中至少参与r个三角形。本文针对任意整数r≥1，从输入图的结构参数视角系统性地研究顶点r-三角形s-团问题。具体而言，我们以输入图的树宽(tw)为参数时，给出了顶点r-三角形2-团的FPT算法；以输入图的h指数为参数时，给出了XP算法。此外，以输入图的反馈边数(fes)为参数时，我们为顶点r-三角形s-团问题给出了规模为O(fes)条边的核。