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A Parameterized Study of Secluded Structures in Directed Graphs

Authors Jonas Schmidt , Shaily Verma , Nadym Mallek

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LIPIcs.ISAAC.2025.53.pdf
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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Secluded Subgraph
  • Parametrized Complexity
  • Directed Graphs
  • Strong Connectivity

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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 problems 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 Π-Subgraph, where given a directed graph G and an integer 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 Out-Secluded ℱ-Free Subgraph problem with parameter k is W[1]-hard, where 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^k n^𝒪(1).

Cite As Get BibTex

Jonas Schmidt, Shaily Verma, and Nadym Mallek. A Parameterized Study of Secluded Structures in Directed Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 53:1-53:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.53

Author Details

Jonas Schmidt
  • Bocconi University, Milan, Italy
Shaily Verma
  • Hasso Plattner Institute, Potsdam, Germany
Nadym Mallek
  • Hasso Plattner Institute, Potsdam, Germany

References

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