Faster Algorithms for Finding Small Induced Patterns in Sparse Host Graphs

2026-09-30

AI Summary

This study addresses the efficient detection of fixed induced subgraphs in sparse graphs. It proposes fast algorithms based on (p,q)-width and pattern polynomials by introducing a generalized notion of tree decomposition width. This approach transcends traditional frameworks that rely solely on width, instead exploiting the structural properties of target patterns for algorithmic design. The method significantly improves upon existing bounds for detecting paths and cycles in connected graphs with five to seven vertices as well as in bipartite graphs. For certain pattern detection tasks, it achieves theoretically optimal time complexity, substantially enhancing the computational efficiency of subgraph matching in sparse graphs.

Abstract

We study algorithms for detecting induced subgraphs corresponding to fixed pattern graphs in host graphs. We show that at least five of the 21 connected graphs on five vertices can be detected in time roughly the product of the number of vertices and the number of edges, and that at least 65 of the 112 connected graphs on six vertices can be detected in time nearly quadratic in the number of edges. We also give algorithms for detecting induced paths and cycles on seven vertices, running in time roughly the number of vertices times the square of the number of edges. Our main technical tool is a generalized notion of tree decomposition width, called (p, q)-width. It yields algorithms whose running times depend on both the number of vertices and the number of edges, and are never worse than existing bounds. Whenever the host graph has fewer than roughly quadratically many edges in its number of vertices, our bounds are strictly faster. For some patterns, including the seven-vertex cycle, our algorithms are optimal under standard complexity-theoretic assumptions. We further develop this approach using pattern-based polynomials that exploit the structure of tree decompositions, not just their width. This gives algorithms for detecting induced paths and cycles on an even number of vertices in bipartite graphs, running in time roughly the (k-1)-th power of the number of edges for paths on 2k vertices, and that same bound times the number of vertices for cycles on 2k vertices. These are faster than the best known algorithms for general graphs.

Problem

• induced subgraph detection
• pattern graphs
• sparse host graphs
• algorithm efficiency

Innovation

• induced subgraph detection
• (p, q)-width
• tree decomposition
• sparse graphs
• pattern-based polynomials