Abstract
Let G be a planar graph whose vertices are colored either red or blue and let S be a set of points having as many red (blue) points as the red (blue) vertices of G. A 2-colored point-set embedding of G on S is a planar drawing that maps each red (blue) vertex of G to a red (blue) point of S. We show that there exist properly 2-colored graphs (i.e., 2-colored graphs with no adjacent vertices having the same color) having treewidth two whose point-set embeddings may require linearly many bends on linearly many edges. For a contrast, we show that two bends per edge are sufficient for 2-colored point-set embedding of properly 2-colored outerplanar graphs. For separable point sets this bound reduces to one, which is worst-case optimal. If the 2-coloring of the outerplanar graph is not proper, three bends per edge are sufficient and one bend per edge (which is worst-case optimal) is sufficient for caterpillars.
Work partially supported by: MIUR, grant 20174LF3T8 AHeAD: efficient Algorithms for HArnessing networked Data, and grant SVV–2020–260578.
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Di Giacomo, E., Hančl, J., Liotta, G. (2021). 2-Colored Point-Set Embeddings of Partial 2-Trees. In: Uehara, R., Hong, SH., Nandy, S.C. (eds) WALCOM: Algorithms and Computation. WALCOM 2021. Lecture Notes in Computer Science(), vol 12635. Springer, Cham. https://doi.org/10.1007/978-3-030-68211-8_20
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