Abstract
Given a host graph G and a pattern graph H, the induced subgraph isomorphism problem is to decide whether G contains an induced subgraph that is isomorphic to H. We study the time complexity of induced subgraph isomorphism problems when the pattern graph is fixed. Nesetril and Poljak gave an O(n^{k omega}) time algorithm that decides the induced subgraph isomorphism problem for any 3k vertex pattern graph (the universal algorithm), where omega is the matrix multiplication exponent. Improvements are not known for any infinite pattern family.
Algorithms faster than the universal algorithm are known only for a finite number of pattern graphs. In this paper, we show that there exists infinitely many pattern graphs for which the induced subgraph isomorphism problem has algorithms faster than the universal algorithm.
Our algorithm works by reducing the pattern detection problem into a multilinear term detection problem on special classes of polynomials called graph pattern polynomials. We show that many of the existing algorithms including the universal algorithm can also be described in terms of such a reduction. We formalize this class of algorithms by defining graph pattern polynomial families and defining a notion of reduction between these polynomial families. The reduction also allows us to argue about relative hardness of various graph pattern detection problems within this framework. We show that solving the induced subgraph isomorphism for any pattern graph that contains a kclique is at least as hard detecting kcliques. An equivalent theorem is not known in the general case.
In the full version of this paper, we obtain new algorithms for P_5 and C_5 that are optimal under reasonable hardness assumptions. We also use this method to derive new combinatorial algorithms  algorithms that do not use fast matrix multiplication  for paths and cycles. We also show why graph homomorphisms play a major role in algorithms for subgraph isomorphism problems. Using this, we show that the arithmetic circuit complexity of the graph homomorphism polynomial for K_k  e (The kclique with an edge removed) is related to the complexity of many subgraph isomorphism problems. This generalizes and unifies many existing results.
BibTeX  Entry
@InProceedings{blser_et_al:LIPIcs:2018:9917,
author = {Markus Bl{\"a}ser and Balagopal Komarath and Karteek Sreenivasaiah},
title = {{Graph Pattern Polynomials}},
booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)},
pages = {18:118:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770934},
ISSN = {18688969},
year = {2018},
volume = {122},
editor = {Sumit Ganguly and Paritosh Pandya},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9917},
URN = {urn:nbn:de:0030drops99172},
doi = {10.4230/LIPIcs.FSTTCS.2018.18},
annote = {Keywords: algorithms, induced subgraph detection, algebraic framework}
}
Keywords: 

algorithms, induced subgraph detection, algebraic framework 
Collection: 

38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018) 
Issue Date: 

2018 
Date of publication: 

05.12.2018 