Boolean combinations of graphs

TL;DR

This work introduces and develops a systematic framework for boolean combinations of graphs, defining how a target graph G can be expressed as a boolean function of several base graphs on the same vertex set. It establishes formal representations (union, intersection, XOR) and extends them to a broad class of graph families, deriving preservation results for speed, labeling schemes, Erdős–Hajnal property, and VC dimension, while highlighting non-preservation for χ-boundedness in general. The authors characterize boolean combinations for several special classes (intersection-closed, monotone, equivalence graphs) and prove nuanced χ-boundedness results, including polynomial bounds for split, permutation, and equivalence-graph families. A model-theoretic perspective situates boolean combinations alongside transductions, outlining how these operations relate to expansions and definability, and proposes directions like boolean dimension to quantify complexity. Overall, the paper lays foundational tools to analyze the expressive power, limitations, and algorithmic consequences of boolean graph representations, with implications for adjacency labeling and structural graph theory.

Abstract

Boolean combinations allow combining given combinatorial objects to obtain new, potentially more complicated, objects. In this paper, we initiate a systematic study of this idea applied to graphs. In order to understand expressive power and limitations of boolean combinations in this context, we investigate how they affect different combinatorial and structural properties of graphs, in particular $χ$-boundedness, as well as characterize the structure of boolean combinations of graphs from various classes.

Figures (1)

• Figure 1: A hierarchy of classes of equivalence graphs with respect to the expressiveness of their boolean functions. Each node in the hierarchy corresponds to a hereditary class of equivalence graphs. For each class in the hierarchy, except $\mathcal{E}_1$, every proper subclass of this class is a boolean function of one the classes immediately below it; the same almost holds for $\mathcal{E}_1$, namely, any proper subclass $\mathcal{E}_1$ becomes a function of $\mathcal{E}_0$ after removing finitely many graphs from it (this exceptional relation is reflected in the picture with the dashed arrow). The text beside a node explains characterization(s) of boolean functions of the corresponding class.

Theorems & Definitions (106)

• Theorem 2.1: see e.g. CH11
• Theorem 2.2: Zhegalkin polynomial aka Algebraic Normal Form Zhe27
• Remark 2.7
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4