Bisimplicial separators

A minimal separator of a graph $G$ is a set $S\subseteq V\left(G\right)$ such that there exist vertices $a,b\in V\left(G\right)⧹S$ 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 $G_k$ the class of all graphs in which each minimal separator is $k$-simplicial. We show that for each $k\ge 0$, the class $G_k$ 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 $G_k$. We also give a complete list of minimal forbidden induced minors for $G_2$. Next, we show that, for $k\ge 1$, every nonnull graph in $G_k$ has a $k$-simplicial vertex, that is, 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 $G_2$. Further, we show that, for $k\ge 3$, it is NP-hard to recognize graphs in $G_k$; the time complexity of recognizing graphs in $G_2$ is unknown. We also show that the  Maximum Clique problem is NP-hard for graphs in $G_3$. Finally, we prove a decomposition theorem for diamond-free graphs in $G_2$ (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 $G_2$, and improved running times for the  Maximum Weight Clique and  Maximum Weight Stable Set problems for this class of graphs.