Counting Patterns in Degenerate Graphs in Constant Space

2025-11-06

AI Summary

This paper investigates constant-space algorithms for counting homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms of a fixed pattern graph in *d*-degenerate graphs. It introduces a novel graph parameter—*DAG-tree depth*—and combines it with DAG-tree width to design efficient divide-and-conquer and dynamic programming algorithms under the unit-cost RAM model. The work achieves the first constant-space subgraph counting algorithm for sparse pattern graphs; provides a complete induced-subgraph characterization for graphs with DAG-tree depth ≤ 2; attains *O*(*n*³) time with constant space for all pattern graphs on at most nine vertices; and reaches *O*(*n*²) time for induced subgraph counting when the pattern has at most eleven vertices. The core innovation lies in establishing DAG-tree depth as a new structural dimension for sparse graphs and proving its tight connection to low-space computational tractability.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

For an arbitrary, fixed graph (pattern graph), we study the algorithmic complexity of counting homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms from the pattern graph to $n$-vertex, $d$-degenerate graphs as input. Recent work by Bressan (Algorithmica, 2021) has shown that this problem has efficient dynamic programming algorithms using a graph parameter called DAG treewidth. Bressan used DAG treewidth to design a fast algorithm for counting homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms that use polynomial space. Bera, Gishboliner, Levanzov, Seshadhri, and Shapira (SODA, 2021) provided a characterization of graphs with DAG treewidth one. In this paper, we introduce a new graph parameter called DAG treedepth and show that it yields efficient divide and conquer algorithms that use only constant space (in the unit-cost RAM model). Specifically, we show: An algorithm for counting subgraphs isomorphic to sparse pattern graphs using only constant space. We derive an induced minor-based characterization for graphs of DAG treedepth up to two. For pattern graphs upto nine vertices, the induced subgraphs can be counted in $O(n^3)$ time using constant space. An algorithm for counting induced subgraphs that matches the running time given by Bressan but only uses constant space. Apart from the DAG treedepth result, we also focus on DAG treewidth. For DAG treewidth, we show that we can count homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms faster than Bressan's algorithm (2021). We further show that for all pattern graphs up to 11 vertices, we can count induced subgraphs in quadratic time.

Problem

• Counting graph patterns in degenerate graphs using constant space
• Developing efficient algorithms for subgraph isomorphism counting problems
• Improving computational complexity for homomorphism and subgraph counting

Innovation

• Introducing DAG treedepth for constant-space algorithms
• Using divide and conquer approach with constant memory
• Counting induced subgraphs in O(n³) time for small patterns