Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels

TL;DR

This paper investigates the complexity of finding specific subgraphs within larger graphs, identifying properties of graph classes that determine whether these problems are efficiently solvable, kernelizable, or hard.

Contribution

It introduces new combinatorial properties that characterize when subgraph problems are tractable or hard for hereditary graph classes.

Findings

Hereditary class F with property X allows polynomial kernels for F-Packing.

Classes lacking property X lead to hardness results or no polynomial Turing kernels.

Specific structural properties determine the complexity and kernelizability of subgraph problems.

Abstract

We study two fundamental problems related to finding subgraphs: (1) given graphs G and H, Subgraph Test asks if H is isomorphic to a subgraph of G, (2) given graphs G, H, and an integer t, Packing asks if G contains t vertex-disjoint subgraphs isomorphic to H. For every graph class F, let F-Subgraph Test and F-Packing be the special cases of the two problems where H is restricted to be in F. Our goal is to study which classes F make the two problems tractable in one of the following senses: * (randomized) polynomial-time solvable, * admits a polynomial (many-one) kernel, or * admits a polynomial Turing kernel (that is, has an adaptive polynomial-time procedure that reduces the problem to a polynomial number of instances, each of which has size bounded polynomially by the size of the solution). We identify a simple combinatorial property such that if a hereditary class F has this property, then F-Packing admits a polynomial kernel, and has no polynomial (many-one) kernel otherwise, unless the polynomial hierarchy collapses. Furthermore, if F does not have this property, then F-Packing is either WK[1]-hard, W[1]-hard, or Long Path-hard, giving evidence that it does not admit polynomial Turing kernels either. For F-Subgraph Test, we show that if every graph of a hereditary class F satisfies the property that it is possible to delete a bounded number of vertices such that every remaining component has size at most two, then F-Subgraph Test is solvable in randomized polynomial time and it is NP-hard otherwise. We introduce a combinatorial property called (a,b,c,d)-splittability and show that if every graph in a hereditary class F has this property, then F-Subgraph Test admits a polynomial Turing kernel and it is WK[1]-hard, W[1]-hard, or Long Path-hard, otherwise.