Parameterized Complexity of Finding a Maximum Common Vertex Subgraph Without Isolated Vertices
Palash Dey, Anubhav Dhar, Ashlesha Hota, Sudeshna Kolay, Aritra Mitra
TL;DR
This work studies Maximum Common Vertex Subgraph (MaxComSubG), the problem of finding a largest common subgraph with no isolated vertices in two input graphs. It proves NP-hardness and provides an FPT algorithm parameterized by the subgraph size $h$, then delivers a comprehensive parameterized complexity map for structural parameters such as vertex cover, treedepth, pathwidth, and treewidth, including an FPT result for the sum of vertex-cover numbers and an EPTAS for planar graphs of bounded degree. The results show a diverse landscape: some parameters yield hardness (e.g., $ ext{W[2]}$-hardness for vertex cover, para-$ ext{NP}$-hardness for treedepth/pathwidth/treewidth and max degree), while others permit tractable algorithms under combined parameters. The EPTAS demonstrates practical approximation in restricted graph classes, and the study’s insights advance understanding of how structural graph properties influence the tractability of subgraph packing problems with isolation constraints.
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.