Parameterized Complexity of Finding a Maximum Common Vertex Subgraph Without Isolated Vertices

TL;DR

This paper investigates the computational complexity of a variant of the Maximum Common Vertex Subgraph problem, establishing NP-hardness, an FPT algorithm, and a comprehensive parameterized complexity classification based on structural graph parameters.

Contribution

It provides the first complexity classification of the problem with respect to multiple structural parameters and their combinations, along with an FPT algorithm for the problem.

Findings

The problem is NP-hard.

An FPT algorithm parameterized by h is developed.

A complete dichotomy of parameterized results for various structural parameters.

Abstract

In this paper, we study the Maximum Common Vertex Subgraph problem: Given two input graphs $G_1,G_2$ and a non-negative integer $h$, is there a common subgraph $H$ on at least $h$ vertices such that there is no isolated vertex in $H$. In other words, each connected component of $H$ has at least $2$ vertices. This problem naturally arises in graph theory along with other variants of the well-studied Maximum Common Subgraph problem and also has applications in computational social choice. We show that this problem is NP-hard and provide an FPT algorithm when parameterized by $h$. Next, we conduct a study of the problem on common structural parameters like vertex cover number, maximum degree, treedepth, pathwidth and treewidth of one or both input graphs. We derive a complete dichotomy of parameterized results for our problem with respect to individual parameterizations as well as combinations of parameterizations from the above structural parameters. This provides us with a deep insight into the complexity theoretic and parameterized landscape of this problem.