将派系变成路径以实现平面性
Turning Cliques into Paths to Achieve Planarity
摘要：
出于混合图表示的动机，我们引入并研究了以下超平面性问题，我们称之为$h$ -C2 P 2路径平面性：给定一个图$G$，其顶点被划分为大小至多为$h$的子集，每个子集诱导一个团，从每个团中删除边，使得由每个子集诱导的子图是路径，以这种方式，得到的子图$G$是平面的。当$G$是简单拓扑图时，我们研究了这个问题，并建立了它的复杂性与$k$ -平面性的关系.本文证明了当$h=4$和$G$是简单的$3$ -平面图时，$h$ -C_2路径平面性是NP-完全的，而当$G$是$1$ -平面图时，对任何$h$，它都能在线性时间内求解.
英文摘要：
Motivated by hybrid graph representations, we introduce and study the following beyond-planarity problem, which we call $h$-Clique2Path Planarity: Given a graph $G$, whose vertices are partitioned into subsets of size at most $h$, each inducing a clique, remove edges from each clique so that the subgraph induced by each subset is a path, in such a way that the resulting subgraph of $G$ is planar. We study this problem when $G$ is a simple topological graph, and establish its complexity in relation to $k$-planarity. We prove that $h$-Clique2Path Planarity is NP-complete even when $h=4$ and $G$ is a simple $3$-plane graph, while it can be solved in linear time, for any $h$, when $G$ is $1$-plane.
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