The complexity of forbidden subgraph sandwich problems and the skew partition sandwich problem

International audienceThe $\Pi$ graph sandwich problem asks, for a pair of graphs$G_1=(V,E_1)$ and $G_2=(V,E_2)$ with $E_1\subseteq E_2$, whether thereexists a graph $G=(V,E)$ that satisfies property $\Pi$ and$E_1\subseteq E \subseteq E_2$. We consider the property of being$F$-free, where $F$ is a fixed graph. We show that the claw-freegraph sandwich and the bull-free graph sandwich problems are bothNP-complete, but the paw-free graph sandwich problem is polynomial.This completes the study of all cases where $F$ has at most fourvertices. A skew partition of a graph $G$ is a partition of itsvertex set into four nonempty parts $A, B, C, D$ such that each vertexof $A$ is adjacent to each vertex of $B$, and each vertex of $C$ isnonadjacent to each vertex of $D$. We prove that the skew partitionsandwich problem is NP-complete, establishing a computationalcomplexity non-monotonicity