Simultaneous Interval Number in Graph Theory

• Simultaneous interval number is a graph parameter that measures the minimal number of label sets required to encode edge structure via interval representations, generalizing standard interval graphs.
• Computing the simultaneous interval number is NP-hard for d ≥ 2, though fixed-parameter tractable algorithms exist under specific modular decompositions and bounded graph parameters.
• Structural bounds relate this parameter to measures like pathwidth and mim-width, offering insights into algorithm optimization and complexity classifications in simultaneous graph representations.

The simultaneous interval number is a structural width parameter for undirected graphs that generalizes interval graphs via interval representations augmented with labeling schemes. It measures the minimal complexity required to encode the edge structure of a graph using intervals on the real line and sets of labels, and it is intimately related to simultaneous representation problems arising in the study of intersection graphs. The parameter arises both as an explicit measure for single graphs and as an implicit threshold in simultaneous representations of families of interval graphs.

1. Definition and Fundamental Properties

A d-simultaneous interval representation of a graph $G=(V,E)$ is an assignment of:

• a real interval $I(v) \subset \mathbb{R}$ for each vertex $v \in V$,
• a label set $L(v) \subseteq \{1,2,\dots,d\}$ for each $v \in V$,

such that $uv \in E(G) \iff [I(u)\cap I(v) \neq \emptyset]$ and $L(u)\cap L(v) \neq \emptyset$. The simultaneous interval number of $G$, denoted $\mathsf{sim}(G)$ or $\mathrm{si}(G)$, is the smallest $d$ for which such a representation exists (Beisegel et al., 2024, Bonomo-Braberman et al., 28 Dec 2025):

$\mathsf{sim}(G) = \min\{\, d \mid G \text{ admits a $d$-simultaneous interval representation}\,\}$

Every interval graph satisfies $\mathsf{sim}(G)=1$. Non-interval graphs such as $C_4$ (the 4-cycle) require $\mathsf{sim}(C_4)=2$. If a family of interval graphs $G_1,\dots,G_k$ admits a simultaneous representation (every shared vertex is modeled by the same interval across all graphs), the minimal $k$ is naturally identified as the simultaneous interval number of the family (Bok et al., 2018).

2. Complexity of Recognition and Computation

Determining whether a graph or graph family has $\mathsf{sim}(G)\leq d$ is $\mathsf{NP}$-hard for $d\geq 2$ (Beisegel et al., 2024, Bonomo-Braberman et al., 28 Dec 2025). This hardness is established by reductions from classical combinatorial problems:

• The simultaneous representation problem for $k$ interval graphs ($\mathsf{SimRep}(\mathsf{Interval})$) is $\mathsf{NP}$-complete in the non-sunflower case when $k$ is part of the input (Bok et al., 2018).
• For a given graph $G$, the decision problem "does $\mathsf{sim}(G)\leq d$?" is $\mathsf{NP}$-hard via reductions from edge-clique-cover and total ordering (Beisegel et al., 2024). Thus, computing $\mathsf{sim}(G)$ is $\mathsf{NP}$-hard.

No efficient algorithms or approximation algorithms for general graphs are known; any such procedures would inherit the intractability of the base problems.

3. Modular Partitions and Parameterized Algorithms

The simultaneous interval number exhibits tractability when parameterized by strong graph decompositions. A key notion is the $\mathcal{G}$-modular cardinality: for a hereditary class $\mathcal{G}$, $\mathcal{G}\textrm{-mc}(G)$ is the smallest cardinality of a partition of $V(G)$ into $\mathcal{G}$-modules. Cluster (union of cliques) and interval graph classes are notable choices (Bonomo-Braberman et al., 28 Dec 2025).

Given a cluster-modular partition of $G$ into $k$ modules, there is a fixed-parameter tractable (FPT) algorithm deciding $\mathsf{sim}(G)\leq d$ in time $O(2^k (4k)! 2^{2kd} n^c)$ (Bonomo-Braberman et al., 28 Dec 2025). The FPT algorithm exploits contraction of clique modules and reduction of independent sets to bounded representatives; it proceeds via branching and enumeration over possible interval and label assignments on a reduced graph of $O(k)$ vertices.

Neighborhood diversity, twin-cover, and vertex cover further bound cluster-modular cardinality, rendering $\mathsf{sim}(G)$ FPT in these parameters plus solution size.

Polynomial kernelization is excluded for $\mathsf{sim}(G)$ when parameterized by treewidth, pathwidth, bandwidth, mim-width, clique-width, modular-width, or even the parameter itself, unless $\mathsf{NP} \subseteq \mathrm{coNP}/\mathrm{poly}$ (Bonomo-Braberman et al., 28 Dec 2025).

4. Structural Bounds and Relationships to Other Parameters

The simultaneous interval number is sandwiched between prominent graph width parameters (Beisegel et al., 2024):

• Lower bound: $\mathit{lmim}(G) \leq \mathsf{sim}(G)$, where $\mathit{lmim}$ is the linear mim-width.
• Upper bound: $$\mathsf{sim}(G) \leq \mathit{pw}(G)^2 + \mathit{pw}(G)$$, where $\mathit{pw}$ is the pathwidth.

Other bounding relationships include:

• $\mathsf{sim}(G) \leq \mathrm{ecc}(G)$ (edge-clique-cover number).
• $\mathrm{thin}(G) \leq 2^{\mathsf{sim}(G)}$ (thinness parameter).

The simultaneous interval number thus interpolates between interval graph structure and more general notions of interval-like representations.

5. Algorithmic Consequences and Problem Complexity

Assuming access to a $d$-simultaneous interval representation, several classic problems become tractable, but boundaries of intractability persist (Beisegel et al., 2024):

• Maximum Clique: At most $2^{2^d} n$ maximal cliques; enumeration and optimization in time $O(d 2^{2^d+2d} n\log n)$.
• Clique of Prescribed Size: Decidable in $O(2^{dk} n)$ time for clique size $k$.
• Independent Set / Dominating Set: FPT algorithms in time $O(2^{dk}n)$, parameterized by solution size $k$ and simultaneous interval number $d$.

Hardness results:

• Graph Coloring: $\mathsf{NP}$-complete for $\mathsf{sim}(G)\leq 2$ even with representation given (Beisegel et al., 2024).
• Independent Dominating Set: W[1]-hard parameterized by $\mathsf{sim}(G)$, with no $f(d) n^{o(d)}$-time algorithm under ETH.

Problems such as Independent Set and Dominating Set admit FPT algorithms, while others (Graph Coloring, Independent Dominating Set) remain computationally hard, sharply distinguishing the simultaneous interval number from other width measures.

6. Illustrative Examples and Sunflower Position

For the 4-cycle $C_4$ (not an interval graph), a $2$-simultaneous interval representation is possible by assigning complementary intervals and label sets to opposite vertex pairs (Beisegel et al., 2024, Bonomo-Braberman et al., 28 Dec 2025). For families of interval graphs, the complexity of simultaneous representation depends on their intersection structure:

• Sunflower position: All pairs of graphs have the same intersection; algorithms are tractable for $k=2$ (linear time), but complexity is unknown for arbitrary $k$ (Bok et al., 2018).
• Non-sunflower position: $\mathsf{SimRep}(\mathsf{Interval})$ is $\mathsf{NP}$-complete when $k$ is part of the input (Bok et al., 2018).

7. Open Problems, Parameter Significance, and Future Directions

The simultaneous interval number quantifies "how far" a graph is from being an interval graph via the minimal number of "tracks" (labels) enabling a single-interval-per-vertex intersection model (Beisegel et al., 2024, Bonomo-Braberman et al., 28 Dec 2025). Key open questions include:

• Existence of FPT or XP algorithms for recognizing $\mathsf{sim}(G)\leq d$ and constructing such representations.
• Whether enumeration bounds for cliques can be improved.
• Identification of further hard problems becoming FPT for parameterization by $\mathsf{sim}(G)$ plus solution size, such as Feedback Vertex Set or Odd Cycle Transversal.

The parameter is strict enough to enable efficient algorithms for various $\mathsf{NP}$-hard problems, yet delicate enough that some classical graph problems remain hard even at low parameter values, e.g., $d=2$. The study of $\mathsf{sim}(G)$ thus advances both structural and algorithmic understanding of graph classes between interval graphs and more general intersection representations (Beisegel et al., 2024, Bonomo-Braberman et al., 28 Dec 2025, Bok et al., 2018).