A Parameterized Study of Secluded Structures in Directed Graphs

2025-02-09

AI Summary

This paper investigates the “hidden Π-subgraph” problem on directed graphs: given a directed graph $G$ and an integer $k$, find a maximum induced subgraph satisfying property $Pi$ whose number of neighbors in the remainder of $G$—measured via in-neighbors, out-neighbors, or total neighbors—is at most $k$. It presents the first systematic parameterized complexity study of hidden subgraphs in directed graphs. The authors introduce three novel models—In-hidden, Out-hidden, and Total-hidden—establishing a theoretical foundation for directed hidden structures. They prove that In-hidden and Out-hidden DAG subgraphs are W[1]-hard with respect to $k$, revealing a fundamental divergence from the undirected case. A unified FPT framework is developed for $alpha$-bounded graphs. A $k$-FPT algorithm is obtained for Total-hidden strongly connected subgraphs. Moreover, In-hidden and Out-hidden $F$-free subgraphs are shown to be W[1]-hard parameterized by $k + w$, where $w$ is the treewidth of $F$. Finally, the algorithm for hidden cliques is improved to $O(1.6181^k n^{O(1)})$.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Given an undirected graph $G$ and an integer $k$, the Secluded $Pi$-Subgraph problem asks you to find a maximum size induced subgraph that satisfies a property $Pi$ and has at most $k$ neighbors in the rest of the graph. This problem has been extensively studied; however, there is no prior study of the problem in directed graphs. This question has been mentioned by Jansen et al. [ISAAC'23]. In this paper, we initiate the study of Secluded Subgraph problem in directed graphs by incorporating different notions of neighborhoods: in-neighborhood, out-neighborhood, and their union. Formally, we call these problems {{In, Out, Total}-Secluded $Pi$-Subgraph, where given a directed graph $G$ and integers $k$, we want to find an induced subgraph satisfying $Pi$ of maximum size that has at most $k$ in/out/total-neighbors in the rest of the graph, respectively. We investigate the parameterized complexity of these problems for different properties $Pi$. In particular, we prove the following parameterized results: - We design an FPT algorithm for the Total-Secluded Strongly Connected Subgraph problem when parameterized by $k$. - We show that the In/Out-Secluded $mathcal{F}$-Free Subgraph problem with parameter $k+w$ is W[1]-hard, where $mathcal{F}$ is a family of directed graphs except any subgraph of a star graph whose edges are directed towards the center. This result also implies that In/Out-Secluded DAG is W[1]-hard, unlike the undirected variants of the two problems, which are FPT. - We design an FPT-algorithm for In/Out/Total-Secluded $alpha$-Bounded Subgraph when parameterized by $k$, where $alpha$-bounded graphs are a superclass of tournaments. - For undirected graphs, we improve the best-known FPT algorithm for Secluded Clique by providing a faster FPT algorithm that runs in time $1.6181^kn^{mathcal{O}(1)}$.

Problem

• Study Secluded Subgraph in directed graphs.
• Incorporate different neighborhood notions: in, out, total.
• Investigate parameterized complexity for various properties.

Innovation

• Initiates Secluded Subgraph study in directed graphs
• Designs FPT algorithm for Total-Secluded Strongly Connected Subgraph
• Improves FPT algorithm for Secluded Clique in undirected graphs