Graph Pattern Polynomials

Finding and Counting Patterns in Sparse Graphs

“…This is also a quadratic improvement over the O(n k−2 ) time algorithm given by Bläser, Komarath, and Sreenivasaiah [2] when the host graph is sparse. These algorithms are obtained by analyzing the matched treewidth and the automorphism structure of a set of graphs derived from the pattern.…”

“…This algorithm is no worse than the O(n 4 ) time algorithm that can be derived using the techniques by Bläser, Komarath, and Sreenivasaiah [2]. For sparse graphs, our algorithm provides a quadratic improvement.…”

“…Indeed, it can be shown that this algorithm can be easily generalized to efficiently construct circuits for homomorphism polynomials instead of counting homomorphisms. Bläser, Komarath and Sreenivasaiah [2] showed that efficient constructions for homomorphism polynomials can even be used to detect induced subgraphs in some cases. They also show that many of the faster induced subgraph detection algorithms, such finding four-node subgraphs by Williams et al [22] and five-node subgraphs by Kowaluk, Lingas, and Lundell [15] can be described as algorithms that efficiently construct these homomorphism polynomials.…”

“…For induced subgraph detection, we have to build arithmetic circuits for homomorphism polynomials of graphs that are related to the pattern. These ideas were used by Bläser, Komarath, and Sreenivasaiah [2], for example to obtain O(n 4 )-time algorithm for induced C 6 detection and O(n k−2 )-time algorithm for induced P k detection. The overall idea is to express the induced subgraph isomorphism polynomial of any k-vertex pattern G as a linear combination of subgraph isomorphism polynomials of all k-vertex super-graphs G ′ of G. Then, by choosing a suitable prime p, and by doing arithmetic modulo p, we can eliminate the harder super-graphs G ′ from the linear combination.…”