Generalizing Roberts' characterization of unit interval graphs

TL;DR

This work studies how Roberts' simple characterization of unit interval graphs extends to $d$-interval graphs. It proves that for any $d\ge 2$, every $K_{1,2d+1}$-free interval graph is a unit $d$-interval graph, and provides a linear-time algorithm to obtain such a representation from a given interval representation; however, the same assumptions do not ensure a disjoint unit $d$-interval representation. The paper further explores the balanced case, showing that the relationship between disjoint and non-disjoint definitions depends on $d$, with a complete equivalence for $d=2$ but not for $d>2$, and it identifies a refined claw-based condition under which disjoint unit representations can be achieved. Additionally, it demonstrates that the Roberts-type generalization fails for disjoint unit $d$-interval graphs via counterexamples (starting at $d=2$), and it maps the inclusion relations among various subclasses (unit, disjoint, balanced) across small and larger values of $d$, highlighting open questions for $d>2$. Overall, the paper clarifies how the choice of $d$-interval definition affects characterization, algorithmic recognition, and class containment.

Abstract

For any natural number $d$, a graph $G$ is a (disjoint) $d$-interval graph if it is the intersection graph of (disjoint) $d$-intervals, the union of $d$ (disjoint) intervals on the real line. Two important subclasses of $d$-interval graphs are unit and balanced $d$-interval graphs (where every interval has unit length or all the intervals associated to a same vertex have the same length, respectively). A celebrated result by Roberts gives a simple characterization of unit interval graphs being exactly claw-free interval graphs. Here, we study the generalization of this characterization for $d$-interval graphs. In particular, we prove that for any $d \geq 2$, if $G$ is a $K_{1,2d+1}$-free interval graph, then $G$ is a unit $d$-interval graph. However, somehow surprisingly, under the same assumptions, $G$ is not always a \emph{disjoint} unit $d$-interval graph. This implies that the class of disjoint unit $d$-interval graphs is strictly included in the class of unit $d$-interval graphs. Finally, we study the relationships between the classes obtained under disjoint and non-disjoint $d$-intervals in the balanced case and show that the classes of disjoint balanced 2-intervals and balanced 2-intervals coincide, but this is no longer true for $d>2$.

Figures (10)

• Figure 1: $K_{1,5}$-free interval graphs are not contained in the class of disjoint unit 2-interval graphs. The class of unit 2-interval graphs is a superclass of disjoint unit 2-interval graphs, and spans the whole intersection of $K_{1,5}$-free and interval graphs.
• Figure 2: Interval $I$ intersects 8 disjoint intervals. In red, the 4-interval returned by the algorithm. Note that if $l(I_2)$ were defined as $l(A_3)$ instead of $l(B_3)$, it would create a forbidden $K_{1,3}$.
• Figure 3: Interval representation of a $K_{1,5}$-free graph that cannot be turned into a disjoint unit 2-interval representation just by "cutting" intervals that intersect more than three pairwise disjoint intervals. In the figure, the intervals in red are all obtained using a natural extension of the algorithm. We can see that in this way, $3_2$ intersects three disjoint intervals: $8_1, 8_2, 11$. The reader can check that no other way of stretching the intervals works if $8_1$ and $8_2$ are required to be disjoint.
• Figure 4: The five other graphs on 14 vertices which are interval and $K_{1,5}$-free but not disjoint unit 2-interval. In dashed red, the edges that differ from the graph that we analyze here.
• Figure 5: One of the 6 graphs with 14 vertices (the one with the fewest edges) which is an interval graph (see \ref{['fig:representation']}) and $K_{1,5}$-free, but not disjoint unit 2-interval.

Theorems & Definitions (10)

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