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.

翻译：本文研究最大公共顶点子图问题：给定两个输入图$G_1,G_2$及非负整数$h$，是否存在至少包含$h$个顶点的公共子图$H$，使得$H$中不存在孤立顶点。换言之，$H$的每个连通分量至少包含$2$个顶点。该问题在图论中自然产生于对经典最大公共子图问题的变体研究，并在计算社会选择领域具有应用价值。我们证明该问题是NP难问题，并给出了以$h$为参数的FPT算法。随后，我们针对单个或两个输入图的结构参数（如顶点覆盖数、最大度、树深、路径宽度和树宽）对该问题进行研究。通过对上述结构参数的独立参数化与组合参数化分析，我们建立了该问题的完整参数化结果二分分类，从而深入揭示了该问题的计算复杂性理论与参数化复杂度图景。