The Graph Motif problem parameterized by the structure of the input graph

TL;DR

This paper investigates the complexity of the Graph Motif problem based on various graph structural parameters, providing new algorithms, lower bounds, and revealing unique hardness results.

Contribution

It systematically analyzes the problem's complexity relative to graph structure, offering new FPT algorithms, kernelization bounds, and a novel hardness result for max leaf number.

Findings

New FPT algorithms for various parameters

Kernelization lower bounds and ETH-based lower bounds

First known W[1]-hardness for max leaf number parameter

Abstract

The Graph Motif problem was introduced in 2006 in the context of biological networks. It consists of deciding whether or not a multiset of colors occurs in a connected subgraph of a vertex-colored graph. Graph Motif has been mostly analyzed from the standpoint of parameterized complexity. The main parameters which came into consideration were the size of the multiset and the number of colors. Though, in the many applications of Graph Motif, the input graph originates from real-life and has structure. Motivated by this prosaic observation, we systematically study its complexity relatively to graph structural parameters. For a wide range of parameters, we give new or improved FPT algorithms, or show that the problem remains intractable. For the FPT cases, we also give some kernelization lower bounds as well as some ETH-based lower bounds on the worst case running time. Interestingly, we establish that Graph Motif is W[1]-hard (while in W[P]) for parameter max leaf number, which is, to the best of our knowledge, the first problem to behave this way.