A Systematic Study of Single-Anchor Logical Gadgets

TL;DR

This work studies single-anchor ladgets for $3$-coloring, introducing a formal framework in which one vertex is fixed to color $0$ and the remaining colors $\,\{1,2\}\$ serve as truth values. It develops core primitives—$\text{MOV}$, $\text{NOT}$, $k$-NOT, $\text{ROT}$, and $\text{ROT}_s$—to construct ladgets that implement Boolean functions under the single-anchor constraint, illustrating a distinct building-block structure from traditional two-anchor SAT reductions. Through an exhaustive search of all non-isomorphic connected graphs on 7–10 vertices, the authors identify minimal ladgets for NAND, OR, AND, XOR (and XNOR) and reveal extreme rarity of such gadgets, notably finding only two non-isomorphic XNOR ladgets among ~29 billion configurations. They also present a straightforward embedding method to realize any 3-coloring ladget with at most two inputs inside $k$-coloring, enabling extension to larger color palettes while preserving the underlying logic.

Abstract

We present a systematic study of logical gadgets for 3-coloring under a single anchor constraint, where only one color representing logical falsehood is fixed to a vertex. We introduce a framework of what we call ladgets (logical gadgets), graph gadgets that implement Boolean functions. Then, we define a set of core gadgets, called primitives, which help identify and analyze the logical behavior of ladgets. Next, we examine the structure of several standard ladgets and present several structural constraints for ladgets. Through an exhaustive search of all non-isomorphic connected graphs up to 10 vertices, we verify all minimal constructions for standard ladgets. Notably, we identify exactly two non-isomorphic minimal XNOR ladgets in approximately 29 billion gadget configurations, highlighting the rarity of gadgets capable of expressing logical behavior. We also present an embedding technique that embeds ladgets with less than 3 inputs from 3-coloring into k-coloring. Our work shows how the single anchor constraint creates a fundamentally different framework from the two anchor gadgets used in SAT reductions.

Figures (16)

• Figure 1: MOV gadget structure
• Figure 2: NOT gadget structure
• Figure 3: k-NOT gadget structure
• Figure 4: ROT gadget structure
• Figure 5: $\text{ROT}_s$ gadget structure

Theorems & Definitions (21)

• Definition 2.1: Single Anchor System
• Theorem 3.1
• proof
• Definition 4.1: Ladget
• Definition 4.2: Minimality
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3