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arXiv · 2502.06048

A Parameterized Study of Secluded Structures in Directed Graphs

Jonas Schmidt·Shaily Verma·Nadym Mallek
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Abstract

Given an undirected graph $G$ and an integer $k$, the Secluded $Π$-Subgraph problem asks you to find a maximum size induced subgraph that satisfies a property $Π$ 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 $\{\text{In}, \text{Out}, \text{Total}\}$-Secluded $Π$-Subgraph, where given a directed graph $G$ and integers $k$, we want to find an induced subgraph satisfying $Π$ 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 $Π$. 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 $α$-Bounded Subgraph when parameterized by $k$, where $α$-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 $1.6181^kn^{\mathcal{O}(1)}$.

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Jonas Schmidt, Shaily Verma, Nadym Mallek. 2025-12-15. A Parameterized Study of Secluded Structures in Directed Graphs. https://doi.org/10.4230/lipics.isaac.2025.53

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