The \emph{Sandwich Problem} (SP) for a graph class $\calC$ 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 $\calC$. 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 coNP, but neither in P nor coNP-complete (unless P = coNP).

翻译：对于图类 $\calC$，其\emph{三明治问题}（SP）是如下计算问题：输入为一对图 $(V,E_1)$ 和 $(V,E_2)$，其中 $E_1\subseteq E_2$，任务是判定是否存在边集 $E$ 满足 $E_1\subseteq E \subseteq E_2$，使得图 $(V,E)$ 属于 $\calC$。本文证明许多 SP 对应于无限 $2$-边着色图 $H$ 的约束满足问题（CSP）。我们进而指出，关于 SP 的若干已知复杂性结果亦可从无限域 CSP 的一般复杂性分类中推导得出，这表明将 CSP 理论应用于 SP 的复杂性分类具有广阔前景。我们通过运用约束满足理论的基本工具，为多个图类的 SP 提出了新的复杂性结果，包括多重图的线图、二部多重图的线图、$K_k$-自由完美图，以及通过禁止有限多个诱导子图（如 $\{I_4,P_4\}$-自由图）所描述的图类，从而解决了 Alvarado、Dantas 和 Rautenbach（2019）提出的一个公开问题。我们还构造了一个属于 coNP 但既不在 P 中也不是 coNP-完全（除非 P = coNP）的图三明治问题。