On Structural Parameterizations for the 2-Club Problem

TL;DR

This paper systematically studies the computational complexity of the 2-Club problem across various structural graph parameters, providing new hardness results and fixed-parameter algorithms relevant to real-world network analysis.

Contribution

It offers a comprehensive classification of 2-Club complexity based on structural parameters, including tight NP-hardness results and fixed-parameter algorithms for specific graph classes.

Findings

NP-hard on graphs close to bipartite, three cliques, and low domination number

Algorithm solves 2-Club in polynomial time for graphs with bounded h-index

Proves W[1]-hardness for h-index and NP-hardness for graphs with constant degeneracy

Abstract

The NP-hard 2-Club problem is, given an undirected graph G=(V,E) and l\in N, to decide whether there is a vertex set S\subseteq V of size at least l such that the induced subgraph G[S] has diameter at most two. We make progress towards a systematic classification of the complexity of 2-Club with respect to a hierarchy of prominent structural graph parameters. First, we present the following tight NP-hardness results: 2-Club is NP-hard on graphs that become bipartite by deleting one vertex, on graphs that can be covered by three cliques, and on graphs with domination number two and diameter three. Then, we consider the parameter h-index of the input graph. This parameter is motivated by real-world instances and the fact that 2-Club is fixed-parameter tractable with respect to the larger parameter maximum degree. We present an algorithm that solves 2-Club in |V|^{f(k)} time with k being the h-index. By showing W[1]-hardness for this parameter, we provide evidence that the above algorithm cannot be improved to a fixed-parameter algorithm. Furthermore, the reduction used for this hardness result can be modified to show that 2-Club is NP-hard if the input graph has constant degeneracy. Finally, we show that 2-Club is fixed-parameter tractable with respect to distance to cographs.