Colored Simultaneous Geometric Embeddings and Universal Pointsets

Brandes, U; Erten, C; Estrella-Balderrama, A; Fowler, JJ; Frati, F; Geyer, M; Gutwenger, C; Hong, SH; Kaufmann, M; Kobourov, SG; Liotta, G; Mutzel, P; Symvonis, A

dc.contributor.author	Brandes, U	en
dc.contributor.author	Erten, C	en
dc.contributor.author	Estrella-Balderrama, A	en
dc.contributor.author	Fowler, JJ	en
dc.contributor.author	Frati, F	en
dc.contributor.author	Geyer, M	en
dc.contributor.author	Gutwenger, C	en
dc.contributor.author	Hong, SH	en
dc.contributor.author	Kaufmann, M	en
dc.contributor.author	Kobourov, SG	en
dc.contributor.author	Liotta, G	en
dc.contributor.author	Mutzel, P	en
dc.contributor.author	Symvonis, A	en
dc.date.accessioned	2014-03-01T02:04:28Z
dc.date.available	2014-03-01T02:04:28Z
dc.date.issued	2011	en
dc.identifier.issn	0178-4617	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/29433
dc.subject	Simultaneous embedding	en
dc.subject	Simultaneous geometric embedding	en
dc.subject	Colored simultaneous embedding	en
dc.subject	Universal pointsets	en
dc.subject	Graph drawing	en
dc.subject.classification	Computer Science, Software Engineering	en
dc.subject.classification	Mathematics, Applied	en
dc.subject.other	PLANAR GRAPHS	en
dc.subject.other	FIXED EDGES	en
dc.subject.other	DRAWINGS	en
dc.subject.other	TREES	en
dc.title	Colored Simultaneous Geometric Embeddings and Universal Pointsets	en
heal.type	journalArticle	en
heal.language	English	en
heal.publicationDate	2011	en
heal.abstract	Universal pointsets can be used for visualizing multiple relationships on the same set of objects or for visualizing dynamic graph processes. In simultaneous geometric embeddings, the same point in the plane is used to represent the same object as a way to preserve the viewer's mental map. In colored simultaneous embeddings this restriction is relaxed, by allowing a given object to map to a subset of points in the plane. Specifically, consider a set of graphs on the same set of n vertices partitioned into k colors. Finding a corresponding set of k-colored points in the plane such that each vertex is mapped to a point of the same color so as to allow a straight-line plane drawing of each graph is the problem of colored simultaneous geometric embedding. For n-vertex paths, we show that there exist universal pointsets of size n, colored with two or three colors. We use this result to construct colored simultaneous geometric embeddings for a 2-colored tree together with any number of 2-colored paths, and more generally, a 2-colored outerplanar graph together with any number of 2-colored paths. For n-vertex trees, we construct small near-universal pointsets for 3-colored caterpillars of size n, 3-colored radius-2 stars of size n+3, and 2-colored spiders of size n. For n-vertex outerplanar graphs, we show that these same universal pointsets also suffice for 3-colored K (3)-caterpillars, 3-colored K (3)-stars, and 2-colored fans, respectively. We also present several negative results, showing that there exist a 2-colored planar graph and pseudo-forest, three 3-colored outerplanar graphs, four 4-colored pseudo-forests, three 5-colored pseudo-forests, five 5-colored paths, two 6-colored biconnected outerplanar graphs, three 6-colored cycles, four 6-colored paths, and three 9-colored paths that cannot be simultaneously embedded.	en
heal.publisher	SPRINGER	en
heal.journalName	ALGORITHMICA	en
dc.identifier.isi	ISI:000289242100005	en
dc.identifier.volume	60	en
dc.identifier.issue	3	en
dc.identifier.spage	569	en
dc.identifier.epage	592	en