A graph G is prismatic if for every triangle T of G, every vertex of G not in T has a unique neighbour in T. The complement of a prismatic graph is called \emph{antiprismatic}. The complexity of colouring antiprismatic graphs is still unknown. Equivalently, the complexity of the clique cover problem in prismatic graphs is not known. Chudnovsky and Seymour gave a full structural description of prismatic graphs. They showed that the class can be divided into two subclasses: the orientable prismatic graphs, and the non-orientable prismatic graphs. We give a polynomial time algorithm that solves the clique cover problem in every non-orientable prismatic graph. It relies on the the structural description and on later work of Javadi and Hajebi. We give a polynomial time algorithm which solves the vertex-disjoint triangles problem for every prismatic graph. It does not rely on the structural description.

中文翻译：

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关于着色反棱镜图的复杂性

如果对于 G 的每个三角形 T，G 的每个不在 T 中的顶点在 T 中都有一个唯一的邻居，则图 G 是棱柱形的。棱柱形图的补称为 \emph {antiprismatic}。着色反棱镜图的复杂性仍然未知。等效地，棱柱图中的集团覆盖问题的复杂性是未知的。Chudnovsky 和 ​​Seymour 给出了棱柱图的完整结构描述。他们表明该类可以分为两个子类：可定向棱柱图和不可定向棱柱图。我们给出了一个多项式时间算法来解决每个不可定向棱柱图中的团覆盖问题。它依赖于 Javadi 和 Hajebi 的结构描述和后来的工作。我们给出了一个多项式时间算法，它解决了每个棱柱图的顶点不相交三角形问题。它不依赖于结构描述。