@article{635332691f0b44c0bce285bfc4795044,

title = "On the curve complexity of 3-colored point-set embeddings",

abstract = "We establish new results on the curve complexity of k-colored point-set embeddings when k=3. We show that there exist 3-colored caterpillars with only three independent edges whose 3-colored point-set embeddings may require [Formula presented] bends on [Formula presented] edges. This settles an open problem by Badent et al. [5] about the curve complexity of point set embeddings of k-colored trees and it extends a lower bound by Pach and Wenger [35] to the case that the graph only has O(1) independent edges. Concerning upper bounds, we prove that any 3-colored path admits a 3-colored point-set embedding with curve complexity at most 4. In addition, we introduce a variant of the k-colored simultaneous embeddability problem and study its relationship with the k-colored point-set embeddability problem.",

keywords = "Graph drawing, Point-set embedding, Simultaneous embedding",

author = "{Di Giacomo}, Emilio and Leszek G{\c a}sieniec and Giuseppe Liotta and Alfredo Navarra",

note = "Funding Information: The work has been supported in part by the European project “Geospatial based Environment for Optimisation Systems Addressing Fire Emergencies” (GEO-SAFE), contract no. H2020-691161 , by MIUR project “AHeAD: efficient Algorithms for HArnessing networked Data” under Grant 20174LF3T8 , by the Network Sciences and Technologies (NeST) initiative at University of Liverpool , by the project: “Algoritmi e sistemi di analisi visuale di reti complesse e di grandi dimensioni” - Ricerca di Base 2018, Dipartimento di Ingegneria dell'Universit{\`a} degli Studi di Perugia”, and by the project “Algorithms and Graphs Drawing” - Ricerca di Base 2017, Dipartimento di Matematica e Informatica dell' Universit{\`a} degli Studi di Perugia . Funding Information: The work has been supported in part by the European project ?Geospatial based Environment for Optimisation Systems Addressing Fire Emergencies? (GEO-SAFE), contract no. H2020-691161, by MIUR project ?AHeAD: efficient Algorithms for HArnessing networked Data? under Grant 20174LF3T8, by the Network Sciences and Technologies (NeST) initiative at University of Liverpool, by the project: ?Algoritmi e sistemi di analisi visuale di reti complesse e di grandi dimensioni? - Ricerca di Base 2018, Dipartimento di Ingegneria dell'Universit? degli Studi di Perugia?, and by the project ?Algorithms and Graphs Drawing? - Ricerca di Base 2017, Dipartimento di Matematica e Informatica dell'Universit? degli Studi di Perugia. Publisher Copyright: {\textcopyright} 2020 Elsevier B.V.",

year = "2020",

month = dec,

day = "18",

doi = "10.1016/j.tcs.2020.09.027",

language = "English (US)",

volume = "846",

pages = "114--140",

journal = "Theoretical Computer Science",

issn = "0304-3975",

publisher = "Elsevier",

}