A minimal separator of a graph $G$ is a set $S \subseteq V(G)$ such that there exist vertices $a,b \in V(G) \setminus S$ with the property that $S$ separates $a$ from $b$ in $G$, but no proper subset of $S$ does. For an integer $k\ge 0$, we say that a minimal separator is $k$-simplicial if it can be covered by $k$ cliques and denote by $\mathcal{G}_k$ the class of all graphs in which each minimal separator is $k$-simplicial. We show that for each $k \geq 0$, the class $\mathcal{G}_k$ is closed under induced minors, and we use this to show that the Maximum Weight Stable Set problem can be solved in polynomial time for $\mathcal{G}_k$. We also give a complete list of minimal forbidden induced minors for $\mathcal{G}_2$. Next, we show that, for $k \geq 1$, every nonnull graph in $\mathcal{G}_k$ has a $k$-simplicial vertex, i.e., a vertex whose neighborhood is a union of $k$ cliques; we deduce that the Maximum Weight Clique problem can be solved in polynomial time for graphs in $\mathcal{G}_2$. Further, we show that, for $k \geq 3$, it is NP-hard to recognize graphs in $\mathcal{G}_k$; the time complexity of recognizing graphs in $\mathcal{G}_2$ is unknown. We also show that the Maximum Clique problem is NP-hard for graphs in $\mathcal{G}_3$. Finally, we prove a decomposition theorem for diamond-free graphs in $\mathcal{G}_2$ (where the diamond is the graph obtained from $K_4$ by deleting one edge), and we use this theorem to obtain polynomial-time algorithms for the Vertex Coloring and recognition problems for diamond-free graphs in $\mathcal{G}_2$, and improved running times for the Maximum Weight Clique and Maximum Weight Stable Set problems for this class of graphs.

翻译：图$G$的极小分离集是一个集合$S \subseteq V(G)$，满足存在顶点$a,b \in V(G) \setminus S$使得$S$在$G$中将$a$与$b$分离，但$S$的任何真子集都不具备此性质。对于整数$k\ge 0$，若一个极小分离集可被$k$个团覆盖，则称其为$k$-单纯形分离集，并记$\mathcal{G}_k$为所有极小分离集均为$k$-单纯形分离集的图族。我们证明对每个$k \geq 0$，族$\mathcal{G}_k$在诱导子式下封闭，并利用这一性质证明最大权独立集问题在$\mathcal{G}_k$上可在多项式时间内求解。同时给出$\mathcal{G}_2$的极小禁止诱导子式完整列表。进一步证明，对于$k \geq 1$，$\mathcal{G}_k$中每个非零图均存在$k$-单纯形顶点（即邻域为$k$个团并集的顶点），由此推出最大权团问题在$\mathcal{G}_2$图上可在多项式时间内求解。此外，我们证明当$k \geq 3$时，识别$\mathcal{G}_k$中的图为NP难问题；而识别$\mathcal{G}_2$图的时间复杂度仍未知。同时证明最大团问题在$\mathcal{G}_3$图中为NP难问题。最后，我们给出$\mathcal{G}_2$中无钻石图（钻石图是从$K_4$删除一条边得到的图）的分解定理，并利用该定理得到$\mathcal{G}_2$中无钻石图的顶点着色与识别问题的多项式时间算法，同时改进了这类图最大权团与最大权独立集问题的运行时间。