A Parameterized Study of Secluded Structures in Directed Graphs

TL;DR

This work studies directed variants of the Secluded Π-Subgraph problem, defining In-, Out-, and Total-seclusion and examining their parameterized complexity with respect to k and combined parameters. We prove NP-hardness for Total-Secluded Strongly Connected Subgraph and design an FPT algorithm with runtime $2^{2^{2^{\mathcal{O}(k^2)}}} n^{\mathcal{O}(1)}$ using recursive understanding, balanced separators, and boundary complementations. We further show W[1]-hardness for In/Out-Secluded F-Free Subgraphs under parameter k+w, while providing FPT results for α-bounded and α-bounded subgraphs, and an improved $1.6181^k$ algorithm for Secluded Clique in the undirected setting. To accomplish this, we introduce boundary complementations and a graph-extension compression framework to manage the search space and preserve structure across recursive calls, advancing the algorithmic toolkit for directed secluded subgraph problems.

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 $1.6181^kn^{\mathcal{O}(1)}$.

Figures (2)

• Figure 1: An illustration of the general recursive understanding algorithm used in \ref{['sec:scc']}. There are two recursive calls in total, highlighted with dashed arrows. As defined later, only vertices inside $B$ are allowed to be in the neighborhood of a solution. $W$ is chosen to be the side of the separation with a smaller intersection with $T$.
• Figure 3: A visualization of a solution in the original graph and a solution in a boundary complementation. Every partial solution in $U$ can be represented by a boundary complementation.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Corollary 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Definition 7: Boundary Complementation
• Lemma 8
• Definition 9: Separation