A CSP approach to Graph Sandwich Problems

The Sandwich Problem (SP) for a graph class $\mathcal{C}$ is the following computational problem. The input is a pair of graphs $(V,E_1)$ and $(V,E_2)$ where $E_1 \subseteq E_2$, and the task is to decide whether there is an edge set $E$ where $E_1 \subseteq E \subseteq E_2$ such that the graph $(V,E)$ belongs to $\mathcal{C}$. In this paper we show that many SPs correspond to the constraint satisfaction problem (CSP) of an infinite 2-edge-coloured graph $H$. We then notice that several known complexity results for SPs also follow from general complexity classifications of infinite-domain CSPs, suggesting a fruitful application of the theory of CSPs to complexity classifications of SPs. We strengthen this evidence by using basic tools from constraint satisfaction theory to propose new complexity results of the SP for several graph classes including line graphs of multigraphs, line graphs of bipartite multigraphs, $K_k$-free perfect graphs, and classes described by forbidding finitely many induced subgraphs, such as $\{I_4,P_4\}$-free graphs, settling an open problem of Alvarado, Dantas, and Rautenbach (2019). We also construct a graph sandwich problem which is in $\mathrm{coNP}$, but neither in $\mathrm{P}$ nor $\mathrm{coNP}$-complete (unless $\mathrm{P}=\mathrm{coNP}$).