Academic Journal
Generalizing Roberts' characterization of unit interval graphs
| Τίτλος: | Generalizing Roberts' characterization of unit interval graphs |
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| Συγγραφείς: | Ardévol Martínez, Virginia, Rizzi, Romeo, Saffidine, Abdallah, Sikora, Florian, Vialette, Stéphane |
| Συνεισφορές: | Virginia Ardévol Martínez and Romeo Rizzi and Abdallah Saffidine and Florian Sikora and Stéphane Vialette, Sikora, Florian |
| Publication Status: | Preprint |
| Στοιχεία εκδότη: | arXiv, 2024. |
| Έτος έκδοσης: | 2024 |
| Θεματικοί όροι: | Unit Interval Graphs, FOS: Computer and information sciences, Mathematics of computing → Graph theory, Discrete Mathematics (cs.DM), Characterization, [INFO] Computer Science [cs], Computer Science - Data Structures and Algorithms, Interval graphs, Multiple Interval Graphs, Data Structures and Algorithms (cs.DS), ddc:004, Computer Science - Discrete Mathematics |
| Περιγραφή: | 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 $d>2$. |
| Τύπος εγγράφου: | Article Conference object |
| Περιγραφή αρχείου: | application/pdf |
| DOI: | 10.48550/arxiv.2404.17872 |
| DOI: | 10.4230/lipics.mfcs.2024.12 |
| Σύνδεσμος πρόσβασης: | http://arxiv.org/abs/2404.17872 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.12 https://hal.science/hal-04698633v1 https://doi.org/10.4230/lipics.mfcs.2024.12 |
| Rights: | arXiv Non-Exclusive Distribution CC BY |
| Αριθμός Καταχώρησης: | edsair.doi.dedup.....f371ca4904f289d9551482d1e3e2994e |
| Βάση Δεδομένων: | OpenAIRE |
| DOI: | 10.48550/arxiv.2404.17872 |
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