Generalizing Roberts' characterization of unit interval graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910426373029888 |
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| author | Martínez, Virginia Ardévol Rizzi, Romeo Saffidine, Abdallah Sikora, Florian Vialette, Stéphane |
| author_facet | Martínez, Virginia Ardévol Rizzi, Romeo Saffidine, Abdallah Sikora, Florian Vialette, Stéphane |
| contents | 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 $d>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17872 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalizing Roberts' characterization of unit interval graphs Martínez, Virginia Ardévol Rizzi, Romeo Saffidine, Abdallah Sikora, Florian Vialette, Stéphane Discrete Mathematics Data Structures and Algorithms 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 $d>2$. |
| title | Generalizing Roberts' characterization of unit interval graphs |
| topic | Discrete Mathematics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2404.17872 |