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

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.

💡 Research Summary

The paper investigates the “Maximum Common Vertex Subgraph without Isolated Vertices” problem (abbreviated MaxComSubG). Given two undirected simple graphs G1 and G2 and a non‑negative integer h, the task is to decide whether there exists a common subgraph H of size at least h such that H contains no isolated vertices, i.e., every connected component of H has at least two vertices. This restriction eliminates the trivial solution consisting of isolated vertices that makes the ordinary Maximum Common Subgraph problem uninteresting, and it also corresponds to a natural model in computational social choice: the “maximum neighbour‑happy house allocation” problem, where agents and houses are represented by two graphs and an allocation is happy if each agent is adjacent to a house that is adjacent to a neighbour in the other graph.

The authors first prove that MaxComSubG is NP‑hard (Theorem 17) by reductions from classic hard problems such as Vertex Cover. They then show that the problem becomes fixed‑parameter tractable (FPT) when the parameter is the desired solution size h. The key observation is that any optimal H can be transformed into a star‑forest (a disjoint union of non‑trivial stars). By enumerating possible choices of star centres and leaves, they obtain an algorithm running in O(2^h·poly(n)) time (Theorem 12).

The core contribution is a comprehensive parameterized complexity landscape with respect to several structural graph parameters: vertex‑cover number τ(G), maximum degree Δ(G), treedepth td(G), pathwidth pw(G), and treewidth tw(G). For each parameter, both individual and combined parameterizations are examined, yielding a complete dichotomy of tractable versus intractable cases, summarized in Table 1.

Key hardness results include:

• W