Computing parameters that generalize interval graphs using restricted modular partitions

TL;DR

The paper studies computing the width-like parameters thinness and the simultaneous interval number under parameterizations defined by the G-modular cardinality, focusing on interval and cluster modular classes. It introduces linear-time procedures for obtaining interval-modular and cluster-modular partitions via modular decomposition, and leverages module-reduction lemmas to derive kernelization and FPT results. A linear kernel is achieved for Thinness parameterized by the interval-modular cardinality, and an FPT algorithm is obtained for Simultaneous Interval Number parameterized by the cluster-modular cardinality plus the solution size, with several corollaries for neighborhood diversity, twin-cover, and vertex cover. The work also establishes strong negative kernelization results for several width parameters, and discusses the relationship between modular-width and linear mim-width, along with open questions for future exploration. Overall, the results advance parameterized approaches to distance-from-interval-graph problems and connect modular partitions to practical kernels and fixed-parameter algorithms.

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.

Figures (7)

• Figure 1: The relationships between the different width parameters that we consider in this work, and some classical ones. Here, an arrow from parameter $A$ to parameter $B$ means that $A$ is bounded by a function on $B$. Most of the relationships were known and appear in diagrams of Milanic-simBelm-mwB-B-M-P-convex-jcssmodular-width. The remaining ones are proved in \ref{['prop:interval-modular-cardinality-parametrization']}, \ref{['prop:neighborhood-partition-is-cluster-modular-partition', 'lemma:twin-cover-hcupn', 'lemma:si-tc']}, and the results in \ref{['sec:modular-cardinality', 'sec:parameterizations']}. To explain the incomparability between thinness and modular-width, between simultaneous interval number and both ${\textsf{interval}}\xspace$-modular cardinality and neighborhood diversity, and between ${\textsf{interval}}\xspace$-modular cardinality and modular-width, observe that cographs have modular-width $2$ and unbounded thinness tesis-diego, and complete bipartite graphs have neighborhood diversity $2$ and unbounded simultaneous interval number Milanic-sim, while paths have simultaneous interval number $1$, ${\textsf{interval}}\xspace$-modular cardinality $1$, and unbounded modular-width. Cographs have bounded linear mim-width, but inspired by the ideas of H-A-R-lin-mim-trees, we can also build graph families of bounded modular-width and unbounded linear mim-width (see \ref{['sec:modw-vs-lmimw']}).
• Figure 2: Example of a graph and its modular decomposition tree.
• Figure 3: The graphs $H_0$, $H_1$, and $H_2$.
• Figure 4: Modular decomposition of the graph $H_2$.
• Figure 5: For some graphs, replacing a module of thinness 2 with another graph of thinness 2 decreases the thinness of the graph. Here, dotted lines denote modules, meaning that every vertex adjacent to the dotted line is adjacent to every vertex inside the dotted line. Graph $G$ was found by running the algorithms by thinness-repo.

Theorems & Definitions (61)

• Definition 1: $\mathcal{G}$-modular cardinality modular-partitions-MFCS23
• Lemma 2: thinness-of-product-graphs
• Lemma 3
• proof
• Theorem 4
• proof
• Lemma 5: Thinness of the union tesis-diego
• Lemma 6: Thinness of the join thinness-of-product-graphs
• Theorem 7
• proof