Computing parameters that generalize interval graphs using restricted modular partitions

TL;DR

This paper explores the complexity of computing generalized interval graph parameters using restricted modular partitions, providing new fixed-parameter tractable algorithms and kernels for various graph classes.

Contribution

It introduces FPT algorithms and linear kernels for thinness and simultaneous interval number based on modular parameters, extending known results to broader graph classes.

Findings

Linear kernel for Thinness parameterized by interval-modular cardinality.

FPT algorithms for Thinness with twin-cover and vertex cover parameters.

FPT algorithms for Simultaneous Interval Number with neighborhood diversity and related parameters.

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 $\not\subseteq$ coNP/poly.