Parameterized Complexity of Finding Subgraphs with Hereditary Properties on Hereditary Graph Classes

TL;DR

This paper studies the parameterized complexity of finding subgraphs with hereditary properties within hereditary graph classes, providing a comprehensive framework for various graph classes and properties.

Contribution

It establishes the parameterized complexity of the problem on multiple hereditary graph classes, including polynomial-time solvability, FPT, and W[1]-completeness results.

Findings

Polynomial-time solvable on co-bipartite graphs for certain properties.

FPT algorithms for planar, bipartite, and triangle-free graphs with specific properties.

W[1]-complete on C4-free, K_{1,4}-free, and unit disk graphs for certain properties.

Abstract

We investigate the parameterized complexity of finding subgraphs with hereditary properties on graphs belonging to a hereditary graph class. Given a graph $G$, a non-trivial hereditary property $\Pi$ and an integer parameter $k$, the general problem $P(G,\Pi,k)$ asks whether there exists $k$ vertices of $G$ that induce a subgraph satisfying property $\Pi$. This problem, $P(G,\Pi,k)$ has been proved to be NP-complete by Lewis and Yannakakis. The parameterized complexity of this problem is shown to be W[1]-complete by Khot and Raman, if $\Pi$ includes all trivial graphs but not all complete graphs and vice versa; and is fixed-parameter tractable (FPT), otherwise. As the problem is W[1]-complete on general graphs when $\Pi$ includes all trivial graphs but not all complete graphs and vice versa, it is natural to further investigate the problem on restricted graph classes. Motivated by this line of research, we study the problem on graphs which also belong to a hereditary graph class and establish a framework which settles the parameterized complexity of the problem for various hereditary graph classes. In particular, we show that: $P(G,\Pi,k)$ is solvable in polynomial time when the graph $G$ is co-bipartite and $\Pi$ is the property of being planar, bipartite or triangle-free (or vice-versa). $P(G,\Pi,k)$ is FPT when the graph $G$ is planar, bipartite or triangle-free and $\Pi$ is the property of being planar, bipartite or triangle-free, or graph $G$ is co-bipartite and $\Pi$ is the property of being co-bipartite. $P(G,\Pi,k)$ is W[1]-complete when the graph $G$ is $C_4$-free, $K_{1,4}$-free or a unit disk graph and $\Pi$ is the property of being either planar or bipartite.