On the Structural Parameterizations of 2-Club with Triangle Constraints

2025-09-19

AI Summary

This paper investigates the Vertex r-Triangle 2-Club problem: given an undirected graph (G) and integers (k,r geq 1), decide whether there exists a vertex subset (S) of size at least (k) such that the induced subgraph (G[S]) has diameter at most 2 and every vertex in (S) belongs to at least (r) triangles. We present the first systematic parameterized study for (r geq 1). Our method integrates tree decompositions, dynamic programming, and kernelization techniques to design three structural parameterization algorithms: an FPT algorithm parameterized by treewidth, an XP algorithm parameterized by the (h)-index, and an (O( ext{fes}))-edge kernel parameterized by feedback edge number. These results achieve theoretical breakthroughs under treewidth, (h)-index, and feedback edge number—three distinct structural parameters—significantly reducing instance size and improving computational efficiency. This work advances the parameterized complexity theory of dense subgraph mining.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Kernel Methods
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

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.

Problem

• Studies Vertex r-Triangle 2-Club parameterized by structural graph properties
• Develops FPT algorithms using treewidth and XP via h-index
• Provides kernelization results based on feedback edge number

Innovation

• FPT algorithms for treewidth parameterization
• XP algorithm for h-index parameterization
• Kernel of O(fes) edges for feedback edge number