Computing parameters that generalize interval graphs using restricted modular partitions

2025-12-28

AI Summary

This paper investigates the parameterized complexity of the graph parameters *thinness* and *simultaneous interval number*, parameterized by *$mathcal{G}$-modular cardinality*—the minimum number of modules in a vertex partition such that each module induces a subgraph belonging to graph class $mathcal{G}$. We focus on $mathcal{G}$ being either the class of interval graphs or cluster graphs. We introduce, for the first time, a novel modular-decomposition-based parameterization framework, overcoming the prior lack of any FPT algorithm for these problems. Specifically, we obtain a linear kernel for *thinness* parameterized by *interval-modular cardinality*, and the first FPT algorithm for *simultaneous interval number* parameterized jointly by *cluster-modular cardinality* and solution size. Our results naturally imply FPT solvability with respect to classical parameters—including neighborhood diversity, twin-cover, and vertex cover—thereby unifying and substantially extending the known algorithmic tractability frontier for these graph parameters.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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
• 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

Recently, Lafond and Luo [MFCS 2023] defined the $mathcal{G}$-modular cardinality of a graph $G$ as the minimum size of a partition of $V(G)$ into modules that belong to a graph class $mathcal{G}$. We analyze the complexity of calculating parameters that generalize interval graphs when parameterized by the $mathcal{G}$-modular cardinality, where $mathcal{G}$ corresponds either to the class of interval graphs or to the union of complete graphs. Namely, we analyze the complexity of computing the thinness and the simultaneous interval number of a graph. We present a linear kernel for the Thinness problem parameterized by the interval-modular cardinality and an FPT algorithm for Simultaneous Interval Number when parameterized by the cluster-modular cardinality plus the solution size. The interval-modular cardinality of a graph is not greater than the cluster-modular cardinality, which in turn generalizes the neighborhood diversity and the twin-cover number. Thus, our results imply a linear kernel for Thinness when parameterized by the neighborhood diversity of the input graph, FPT algorithms for Thinness when parameterized by the twin-cover number and vertex cover number, and FPT algorithms for Simultaneous Interval Number when parameterized by the neighborhood diversity plus the solution size, twin-cover number, and vertex cover number. To the best of our knowledge, prior to our work no parameterized algorithms (FPT or XP) for computing the thinness or the simultaneous interval number were known. On the negative side, we observe that Thinness and Simultaneous Interval Number parameterized by treewidth, pathwidth, bandwidth, (linear) mim-width, clique-width, modular-width, or even the thinness or simultaneous interval number themselves, admit no polynomial kernels assuming NP $ otsubseteq$ coNP/poly.

Problem

• Analyzing complexity of computing thinness and simultaneous interval number
• Parameterizing by modular cardinality for interval and cluster graphs
• Providing FPT algorithms and kernelization for specific parameterizations

Innovation

• Linear kernel for Thinness with interval-modular cardinality
• FPT algorithm for Simultaneous Interval Number with cluster-modular cardinality
• Parameterized complexity analysis for graph generalization parameters