Flips and Merge-Width in Sparse Graphs

2026-02-13

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

A flip of a graph is obtained by complementing the edge relation within a set of vertices. Flips are typically used to separate vertices in a graph, by increasing the distances between them. We show that in $K_{t,t}$-free graphs, every short sequence of flips can be simulated by a short sequence of vertex deletions that achieves a similar degree of separation: distances in the resulting graph are, up to a factor of three, at least as large as those obtained after the flips. This result provides a simple and uniform explanation of an emerging pattern in structural graph theory and finite model theory: the $K_{t,t}$-free fragment of a tameness notion for dense graphs often coincides with a tameness notion for sparse graphs. As immediate applications, we recover the following known equivalences. In the $K_{t,t}$-free setting, the dense notions (1) bounded shrub-depth, (2) bounded clique-width, (3) bounded flip-width, (4) monadic dependence, respectively, coincide with the sparse notions (1) bounded tree-depth, (2) bounded tree-width, (3) bounded expansion, and (4) no-where dense-ness. Furthermore, we reprove the result by Dreier and Toru\'nczyk (STOC 2025) stating that $K_{t,t}$-free classes of bounded merge-width have bounded expansion. Our proof provides explicit bounds and is direct, as it shows how to construct strong coloring orders (witnesses of bounded expansion) from merge sequences (witnesses of bounded merge-width). Along the way, we identify a new family of graph parameters, dubbed separation-width, that is sandwiched between the strong and weak coloring numbers, and is closely related to the merge-width parameters. We provide evidence that this family of graph parameters, apparently overlooked in the literature, may play a fundamental role in the study of sparse graphs.

Problem

• K_{t,t}-free graphs
• tameness notions
• sparse graphs
• dense graphs
• structural graph theory

Innovation

• flip-width
• merge-width
• separation-width
• K_{t,t}-free graphs
• bounded expansion