The Simultaneous Interval Number: A New Width Parameter that Measures the Similarity to Interval Graphs

TL;DR

The paper introduces the simultaneous interval number $si(G)$ as the smallest $d$ such that a graph admits a $d$-simultaneous interval representation, generalizing interval graphs via label-aware interval models. It establishes NP-hardness of computing $si(G)$ and situates $si$ between pathwidth and linear mim-width, giving $si(G) le pw(G)^2+pw(G)$ and $pw(G) le si(G) \,\omega(G)-1$, with $si(G) le ecc(G)$ as a consequence of an edge-clique-cover framework. For graph classes with bounded $si(G)$ and a fixed number of labels, the work yields FPT algorithms for clique, Independent Set, and Dominating Set, along with hardness results for Independent Dominating Set and Coloring, highlighting practical algorithmic benefits in an interval-graph–like regime. The results position $si(G)$ as a meaningful, interpretable width parameter that captures interval-graph structure and enables tractable parameterized algorithms beyond what mim-width-based approaches offer.

Abstract

We propose a novel way of generalizing the class of interval graphs, via a graph width parameter called the simultaneous interval number. This parameter is related to the simultaneous representation problem for interval graphs and defined as the smallest number $d$ of labels such that the graph admits a $d$-simultaneous interval representation, that is, an assignment of intervals and label sets to the vertices such that two vertices are adjacent if and only if the corresponding intervals, as well as their label sets, intersect. We show that this parameter is $\mathsf{NP}$-hard to compute and give several bounds for the parameter, showing in particular that it is sandwiched between pathwidth and linear mim-width. For classes of graphs with bounded parameter values, assuming that the graph is equipped with a simultaneous interval representation with a constant number of labels, we give $\mathsf{FPT}$ algorithms for the clique, independent set, and dominating set problems, and hardness results for the independent dominating set and coloring problems. The $\mathsf{FPT}$ results for independent set and dominating set are for the simultaneous interval number plus solution size. In contrast, both problems are known to be $\mathsf{W}[1]$-hard for linear mim-width plus solution size.

Figures (2)

• Figure 1: Two forbidden induced subgraphs of interval graphs with $2$-simultaneous interval representations. Yellow intervals have label set $\{1\}$, blue intervals have label set $\{2\}$ and black intervals have label set $\{1,2\}$. Note that the representation of the 4-cycle can be extended to a $2$-simultaneous interval representation of cycles of arbitrary length.
• Figure 2: Diagram illustrating the relations between different graph width parameters. A directed edge from parameter $P$ to parameter $Q$ means that a bounded value of $P$ implies a bounded value for $Q$. If a directed path from $P$ to $Q$ is missing, then parameter $Q$ is unbounded for the graphs of bounded $P$.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Theorem 5
• Definition 6
• Corollary 7
• Theorem 8
• Theorem 9
• Lemma 10
• Theorem 11
• Definition 12: Thinness