Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes

2026-07-12

AI Summary

This work investigates structural bounds for the class of monadic dependence graphs in the context of parameterized first-order model checking. By integrating graph interpretation theory, neighborhood complexity analysis, and a merge-width decomposition framework, it establishes—for the first time—a connection between this graph class and approximately bounded merge width. The main contributions include proving that monadic dependence graphs exhibit near-linear neighborhood complexity, establishing an upper bound of \(n^{o(1)}\) on their radius-1 merge width, and designing an \(O(n^5)\)-time algorithm that constructs a decomposition sequence with merge width \(O(n^{1-1/d} \log n)\). These results resolve a key conjecture for the radius-1 case and provide the first decomposition-based structural characterization of this graph class.

Abstract

Monadic dependence is a proposed structural dividing line for fixed-parameter tractability of first-order model checking on hereditary graph classes. A graph class is \emph{monadically dependent} if the class of all graphs cannot be interpreted in its vertex-colored members using a fixed first-order formula. We prove two structural consequences of monadic dependence. First, every monadically dependent class has \emph{almost linear neighborhood complexity}: for every graph $G$ in the class and every set $A\subseteq V(G)$, the family $\{N_G(v)\cap A : v\in V(G)\}$ has size $|A|^{1+o(1)}$. Second, every $n$-vertex graph in a monadically dependent class has radius-1 merge-width $n^{o(1)}$. Here, merge-width is the decomposition parameter of Dreier and Toruńczyk based on construction sequences; its radius-$r$ version measures local reachability among parts through already resolved pairs. This settles the radius-1 case of the conjectured connection between monadic dependence and almost bounded merge-width and provides the first decomposition-based structural description of monadically dependent graph classes. Our proof is algorithmic: we give an $\mathcal{O}(n^5)$-time algorithm that, given an $n$-vertex graph $G$ such that $|\{N_G(v)\cap A : v\in V(G)\}|\le O(|A|^d)$ for every $A\subseteq V(G)$, computes a construction sequence witnessing radius-1 merge-width $\mathcal{O}(n^{1-1/d}\log n)$.

Problem

• monadic dependence
• neighborhood complexity
• merge-width
• graph classes
• structural graph theory

Innovation

• monadic dependence
• neighborhood complexity
• merge-width
• first-order model checking
• graph decomposition