---
title: Interval-Modular Cardinality
url: https://www.emergentmind.com/topics/interval-modular-cardinality
type: topic
---

# Interval-Modular Cardinality

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 [2307.02021, 2512.22975].

| Parameter  | Comparative Relation with $\mathrm{imc}(G)$ | Reference                   |
|------------|---------------------------------------------|-----------------------------|
| Cluster-modular cardinality ($\mathrm{mc}^{\text{cluster}}$) | $\mathrm{imc}(G) \leq \mathrm{mc}^{\text{cluster}}(G)$ | [2512.22975]                |
| Neighborhood diversity ($\mathrm{nd}$)                   | $\mathrm{mc}^{\text{cluster}}(G) \leq \mathrm{nd}(G)$  | [2512.22975]                |
| Modular-width ($\mathrm{mw}$)                            | $\mathrm{imc}(G) \leq \mathrm{mw}(G)$                  | [2307.02021]                |
| Vertex/twin cover ($\mathrm{vc},\mathrm{tc}$)            | Functional bound via $2^{\mathrm{vc}(G)}+\mathrm{vc}(G)$ | [2512.22975]               |

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 [2512.22975].

## 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      | [2512.22975]     |
| Simult. Interval Number| $\mathrm{mc}^{\text{cluster}}(G) + d$ | FPT                 | [2512.22975]     |

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 $\mathrm{NP} \subseteq \mathrm{coNP}/\mathrm{poly}$ [2512.22975].

## 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 [2512.22975].

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 [2307.02021] and generalized to encompass algorithmic applications for thinness and simultaneous interval number [2512.22975]. 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.

## 7. References

- Bonomo-Braberman, M., Brandwein, P., Sau, I. "Computing parameters that generalize interval graphs using restricted modular partitions" [2512.22975]
- Lafond, M., Luo, J. "Parameterized Complexity of Domination Problems Using Restricted Modular Partitions" [2307.02021]

Source: https://www.emergentmind.com/topics/interval-modular-cardinality