A CSP approach to Graph Sandwich Problems

TL;DR

This work establishes a deep link between graph sandwich problems (SP) and (infinite-domain) constraint satisfaction problems (CSP) by showing that SP$(\mathcal{C})$ often equals CSP$(H^*)$ for an appropriate countable 2-edge-coloured graph $H$ when $\mathcal{C}$ is hereditary, has the joint embedding property, and is closed under split blow-ups. This unifies many known SP classifications and yields new results across graph classes (e.g., line graphs, $K_k$-free perfect graphs, $(p,q)$-split graphs, and permutation graphs) by invoking pp-constructions, polymorphisms, and reductions to finite-domain CSPs. The paper demonstrates NP-hardness, NP-completeness, and even coNP-intermediate phenomena (including a coNP-intermediate SP) within this CSP framework, highlighting both tractable and intractable regimes under a common methodology. Overall, the CSP lens provides a systematic approach to classifying SP complexity and suggests further avenues in infinite-domain CSP theory for graph-theoretic problems with structural restrictions.

Abstract

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).

Figures (4)

• Figure 1: Consider the primitive positive definition (of $\{E\}$ in $\{E\}$) where $\delta_E(x,y):=\exists z,w\; ( E(x,z)\land E(z,w)\land E(w,y) )$. Here, we depict $C_5$ and its pp-power $\Pi(C_5)\cong K_5$.
• Figure 2: A picture for the pp $\delta_{\sim}$ from the proof of Lemma \ref{['lem:12-split']}, where the filled vertices represent existentially quantified variables.
• Figure 3: To the left, a depiction of the blue (solid) and red (dashed) neighbourhoods of a vertex $v$ in the CSP template $(\mathbb Q^2, B, R)$ that describes the SP for permutation graphs. To the right, a depiction of witnesses $w_1,w_2$$(x,y,z)\in R$ whenever $x$ and $z$ are the bottom left and top right corners of the rectangle $S(x,z)$ and $y$ lies inside $S(x,z)$.
• Figure 4: To the left, an illustration of witnesses for $w_1,w_2,w_3,w_4$ showing that if the edges $x_1x_2$ and $x_1x_2$ are both vertical laying in different vertical lines, then $\mathop{\mathrm{Grid}}\nolimits^\ast\models \gamma(x_1,x_2,y_1,y_2)$. Similarly, in the middle an illustration that $\mathop{\mathrm{Grid}}\nolimits^\ast\models \gamma(x_1,x_2,y_1,y_2)$ if $x_1x_2$ and $y_1y_2$ are vertical edges on the same vertical line. (By rotating the gadgets ninety degrees, we obtain similar pictures showing that $\mathop{\mathrm{Grid}}\nolimits^\ast\models\gamma(x_1,x_2,y_1,y_2)$ when $x_1x_2$ and $y_1y_2$ are both horizontal blue edges). To the right, a depiction showing that $\mathop{\mathrm{Grid}}\nolimits^\ast\models \gamma(x_1,x_2,y_1,y_2)$ if $x_1,x_2$ and $y_2,y_2$ are pairs of non-equal non-adjacent vertices.

Theorems & Definitions (48)

• Example 1
• Example 2
• Lemma 1: Corollary 3.5 in wonderland
• Definition 1
• Remark 1
• Lemma 2
• proof
• Proposition 3
• proof
• Corollary 4