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 $G_1, G_2$ 和一个非负整数 $h$，是否存在一个包含至少 $h$ 个顶点的公共子图 $H$，且 $H$ 中无孤立顶点。换言之，$H$ 的每个连通分量至少包含 $2$ 个顶点。该问题自然产生于图论中，与已被广泛研究的最大公共子图问题的其他变体相关，并在计算社会选择学中具有应用。我们证明该问题是 NP-难的，并给出一个以 $h$ 为参数的 FPT 算法。随后，我们基于常见结构参数（如顶点覆盖数、最大度、树深度、路径宽度及树宽度）对单个或两个输入图进行了问题研究。针对上述结构参数的单独参数化及组合参数化，我们推导出问题参数化结果的完全二分性。这为我们深入理解该问题的复杂度理论和参数化格局提供了重要见解。