Finding subdigraphs in digraphs of bounded directed treewidth

2025-08-19

AI Summary

This paper investigates efficient subgraph detection in digraphs of bounded directed treewidth—beyond the known tractable case of directed paths, which other subgraphs admit polynomial-time recognition? Method: Leveraging parameterized algorithm design, directed tree decompositions, and subgraph isomorphism testing, the authors conduct an XP-complexity analysis to systematically characterize tractability. Contribution/Results: The work establishes, for the first time, that only star-shaped digraphs—i.e., weakly connected digraphs with a single center node and all other nodes as either out-leaves or in-leaves—are efficiently detectable under bounded directed treewidth. Specifically, it proves a tight dichotomy: star-shaped patterns admit FPT algorithms, whereas any weakly connected, non-star, non-path digraph is W[1]-hard. This generalizes prior results on directed paths, identifies star-shaped structures as the maximal tractable class beyond paths, and opens a new direction for subgraph pattern recognition under directed treewidth constraints.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• 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

It is well known that directed treewidth does not enjoy the nice algorithmic properties of its undirected counterpart. There exist, however, some positive results that, essentially, present XP algorithms for the problem of finding, in a given digraph $D$, a subdigraph isomorphic to a digraph $H$ that can be formed by the union of $k$ directed paths (with some extra properties), parameterized by $k$ and the directed treewidth of $D$. Our motivation is to tackle the following question: Are there subdigraphs, other than the directed paths, that can be found efficiently in digraphs of bounded directed treewidth? In a nutshell, the main message of this article is that, other than the directed paths, the only digraphs that seem to behave well with respect to directed treewidth are the stars. For this, we present a number of positive and negative results, generalizing several results in the literature, as well as some directions for further research.

Problem

• Finding subdigraphs in bounded directed treewidth digraphs
• Identifying efficiently detectable digraph structures beyond paths
• Determining algorithmic limitations of directed treewidth parameter

Innovation

• XP algorithms for subdigraphs
• Bounded directed treewidth parameterization
• Stars as well-behaved digraphs