Interval-Modular Cardinality

• Interval-modular cardinality is a graph parameter that measures the minimal partitioning of a vertex set into modules each inducing an interval graph.
• It bridges key graph invariants such as cluster-modular cardinality, neighborhood diversity, and modular-width, and is computable in linear time via modular decomposition.
• Applications include efficient kernelization and fixed-parameter tractable algorithms for problems like thinness and simultaneous interval number, enhancing practical graph analysis.

Interval-modular cardinality is a graph parameter that quantifies the minimal partitioning of a graph’s vertex set into modules, each inducing an interval graph. This construct serves as a unifying and intermediate measure within the landscape of structural graph parameters lying between neighborhood diversity, cluster-modular cardinality, and broader width parameters such as modular-width. Interval-modular cardinality is foundational for algorithmic meta-theorems and kernelization results for problems connected to interval graph generalizations, notably thinness and simultaneous interval number.

1. Formal Definitions and Basic Properties

For any graph $G=(V,E)$, a set $M \subseteq V$ is a module if all vertices outside $M$ are either adjacent to every vertex in $M$ or to none. For a (hereditary) graph class $\mathcal{G}$ containing the trivial graph $K_1$, a module $M$ is a $\mathcal{G}$-module if the induced subgraph $G[M] \in \mathcal{G}$. A partition $\mathcal{M} = \{M_1, \dots, M_k\}$ of $V$ is a $\mathcal{G}$-modular partition if each $M_i$ is a $\mathcal{G}$-module. The $\mathcal{G}$-modular cardinality of $G$ is the minimum $k$ for which such a partition of size $k$ exists, denoted $\mathrm{mc}_{\mathcal{G}}(G)$.

Specializing to $\mathcal{G}$ as the class of interval graphs, the interval-modular cardinality of $G$, denoted $\mathrm{imc}(G)$, is the minimum size of a partition of $V$ into modules, each inducing an interval graph. The formal definition:

$$\mathrm{imc}(G) := \min\{ k : \exists\ \text{interval-modular partition of}\ G\ \text{of size}\ k \}$$

This parameter is always well-defined, as partitioning into singletons (each a module inducing a single vertex, i.e., a trivial interval graph) is always possible.

2. Relationship to Other Graph Parameters

Interval-modular cardinality interpolates between several established graph invariants:

• $$\mathrm{imc}(G) \leq \mathrm{mc}^{\text{cluster}}(G)$$, where $\mathcal{G}$ is the class of cluster graphs (disjoint unions of cliques).
• $$\mathrm{mc}^{\text{cluster}}(G) \leq \mathrm{nd}(G)$$, the neighborhood diversity of $G$.
• $$\mathrm{mc}^{\text{cluster}}(G) \leq 2^{\mathrm{tc}(G)} + \mathrm{tc}(G)$$, where $\mathrm{tc}(G)$ is the twin-cover number; the same functional bound applies for the vertex cover number, giving $$\mathrm{imc}(G) \leq 2^{\mathrm{vc}(G)} + \mathrm{vc}(G)$$.

By construction, $\mathrm{imc}(G) \leq \mathrm{mw}(G)$, the modular-width of $G$. Conversely, $\mathrm{imc}(G) \geq \mathrm{nd}(G)$, provided interval graphs contain all edgeless and complete graphs (Lafond et al., 2023, Bonomo-Braberman et al., 28 Dec 2025).

Parameter	Comparative Relation with $\mathrm{imc}(G)$	Reference
Cluster-modular cardinality ($\mathrm{mc}^{\text{cluster}}$)	$$\mathrm{imc}(G) \leq \mathrm{mc}^{\text{cluster}}(G)$$	(Bonomo-Braberman et al., 28 Dec 2025)
Neighborhood diversity ($\mathrm{nd}$)	$$\mathrm{mc}^{\text{cluster}}(G) \leq \mathrm{nd}(G)$$	(Bonomo-Braberman et al., 28 Dec 2025)
Modular-width ($\mathrm{mw}$)	$\mathrm{imc}(G) \leq \mathrm{mw}(G)$	(Lafond et al., 2023)
Vertex/twin cover ($\mathrm{vc},\mathrm{tc}$)	Functional bound via $2^{\mathrm{vc}(G)}+\mathrm{vc}(G)$	(Bonomo-Braberman et al., 28 Dec 2025)

No inequalities below $\mathrm{imc}(G)$ and clique-width, treewidth, pathwidth, or related width measures are established specifically for the interval case.

3. Algorithmic Computation

$\mathrm{imc}(G)$ can be computed in linear time $O(n+m)$ using modular decomposition. The computation proceeds as follows:

• Compute the modular decomposition tree of $G$ in $O(n+m)$ time.
• At each parallel node, collect all child modules inducing interval graphs into a single module, and process the rest recursively.
• At a series node, combine all complete-graph children and at most one noncomplete interval-graph child into one module.
• At a prime node, test if the quotient graph is interval via a constant-size replacement and standard interval graph recognition (linear time).

This ensures that for $G$, an optimal partition into interval modules (and thus $\mathrm{imc}(G)$) is computed in overall linear time (Bonomo-Braberman et al., 28 Dec 2025).

4. Applications: Thinness and Simultaneous Interval Number

The parameter $\mathrm{imc}(G)$ acts as an effective measure for kernelization and fixed-parameter tractable (FPT) algorithms for problems that generalize interval graph recognition, specifically thinness and simultaneous interval number:

• Thinness: $G$ is $k$-thin if there exists a $k$-partition and vertex order fulfilling a specific adjacency propagation property. It holds that $\mathrm{thin}(G) \leq 2\ \mathrm{imc}(G)$. Furthermore, Thinness parameterized by $\mathrm{imc}(G)$ admits a linear-vertex kernel, retaining at most $2k$ (with $k = \mathrm{imc}(G)$) vertices after appropriate contraction and representative selection in each module. Thinness thus also admits linear (or better) kernels parameterized by cluster-modular cardinality, neighborhood diversity, twin-cover, or vertex cover.
• Simultaneous Interval Number: For $G$, the minimal $d$ such that there is a $d$-simultaneous interval representation; parameterization by cluster-modular cardinality allows an FPT algorithm with running time $2^k\,(4k)!\,2^{d^2k}\,\mathrm{poly}(n)$, with $k = \mathrm{mc}^{\text{cluster}}(G)$. The critical step is reducing each module to either one or two representatives, depending on intersection properties, followed by exhaustive enumeration.

A summary of parameterized complexity results is:

Problem	Parameter	Result	Reference
Thinness	$\mathrm{imc}(G)$	Linear kernel, FPT	(Bonomo-Braberman et al., 28 Dec 2025)
Simult. Interval Number	$\mathrm{mc}^{\text{cluster}}(G) + d$	FPT	(Bonomo-Braberman et al., 28 Dec 2025)

Notably, no polynomial kernel exists for Thinness or Simultaneous Interval Number parameterized by treewidth, pathwidth, bandwidth, (linear) mim-width, clique-width, modular-width, or the parameter itself, unless $\mathrm{NP} \subseteq \mathrm{coNP}/\mathrm{poly}$ (Bonomo-Braberman et al., 28 Dec 2025).

5. Structural and Kernelization Lower Bounds

Interval-modular cardinality delineates the effectiveness of kernelization and compression techniques. Specifically, for any parameter $p(G)$ bounded above by $|V|$ or $|E|$ and subadditive under disjoint union, neither Thinness nor Simultaneous Interval Number admits a polynomial kernel parameterized by $p$ unless $\mathrm{NP}\subseteq\mathrm{coNP}/\mathrm{poly}$, as shown via AND-cross-compositions, adapted to this context (Bonomo-Braberman et al., 28 Dec 2025).

The hierarchy of structural parameters situates interval-modular cardinality strictly intermediate between cluster-modular cardinality and modular-width, with its power for kernelization subsumed for parameters that rapidly grow with vertex or edge cover, but not with the more "global" structural measures like treewidth or clique-width.

6. Context and Research Directions

Interval-modular cardinality was independently defined in the context of parameterized complexity of domination problems using restricted modular partitions (Lafond et al., 2023) and generalized to encompass algorithmic applications for thinness and simultaneous interval number (Bonomo-Braberman et al., 28 Dec 2025). The parameter is now established as both efficiently computable and algorithmically powerful for various structural and recognition problems linked to interval-like graph classes.

Current research directions include the detailed study of $\mathcal{G}$-modular cardinalities for additional hereditary classes beyond interval or cluster graphs, as well as further investigation into the boundary between tractability and intractability for graph invariants parameterized by interval-modular cardinality. A plausible implication is that deeper structural analogies to modular-width and split decompositions may produce new algorithmic paradigms for large-scale graph data.

• Bonomo-Braberman, M., Brandwein, P., Sau, I. "Computing parameters that generalize interval graphs using restricted modular partitions" (Bonomo-Braberman et al., 28 Dec 2025)
• Lafond, M., Luo, J. "Parameterized Complexity of Domination Problems Using Restricted Modular Partitions" (Lafond et al., 2023)