Moderately beyond clique-width: reduced component max-leaf and related parameters

2026-04-21

AI Summary

This work proposes a new graph parameter, the maximum leaf number of contracted connected components (denoted cml↓), which strictly lies between clique-width and reduced bandwidth. The parameter is introduced to unify the algorithmic tractability of bounded-clique-width graphs and unit interval graphs. Within the framework of contraction sequences, the authors show that cml↓ is bounded on unit interval graphs but unbounded on planar graphs. They establish a connection between maximum degree and treewidth in sparse graphs with bounded cml↓ and leverage balanced separators together with first-order transductions to design polynomial-time algorithms for NP-hard problems such as Maximum Induced d-Regular Subgraph. A key contribution is the proof that bounded maximum degree in sparse graphs of bounded cml↓ implies bounded treewidth, along with the demonstration that three-dimensional grids have unbounded reduced bandwidth and thus are not first-order transductions of planar graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Reduced parameters [BKW, JCTB '26; BKRT, SODA '22] are defined via contraction sequences. Based on this framework, we introduce the reduced component max-leaf, denoted by $\operatorname{cml}^\downarrow$, where component max-leaf is the maximum number of leaves in any spanning tree of any connected component. Reduced component max-leaf is strictly sandwiched between clique-width and reduced bandwidth, it is bounded in unit interval graphs, and unbounded in planar graphs. We design polynomial-time algorithms for problems such as \textsc{Maximum Induced $d$-Regular Subgraph} and \textsc{Induced Disjoint Paths} in graphs given with a contraction sequence witnessing low $\operatorname{cml}^\downarrow$, unifying and extending tractability results for classes of bounded clique-width and unit interval graphs. We get the following collapses in sparse classes of bounded $\operatorname{cml}^\downarrow$: bounded maximum degree implies bounded treewidth, whereas $K_{t,t}$-subgraph-freeness implies strongly sublinear treewidth; we show the latter, more generally, for classes of bounded reduced cutwidth. We establish the former result by showing that graphs with bounded $\operatorname{cml}^\downarrow$ admit balanced separators dominated by a bounded number of vertices. We then showcase an application of the reduced parameters to establishing non-transducibility results. We prove that for most reduced parameters $p^\downarrow$ (including reduced bandwidth), the family of classes of bounded $p^\downarrow$ is closed under first-order transductions. We then answer a question of [BKW '26] by showing that the 3-dimensional grids have unbounded reduced bandwidth. As the class of planar graphs (or any class of bounded genus) has bounded reduced bandwidth [BKW '26], this reproves a recent result [GPP, LICS '25] that planar graphs do not first-order transduce the 3-dimensional grids.

Problem

• clique-width
• reduced parameters
• component max-leaf
• contraction sequences
• first-order transductions

Innovation

• reduced component max-leaf
• contraction sequences
• clique-width
• first-order transductions
• polynomial-time algorithms