How to Embed a Path onto Two Sets of Points

Abstract

Let R and B be two sets of points such that the points of R are colored red and the points of B are colored blue. Let P be a path such that |R| vertices of P are red and |B| vertices of P are blue. We study the problem of computing a crossing-free drawing of P such that each blue vertex is represented as a point of B and each red vertex of P is represented as a point of R. We show that such a drawing can always be realized by using at most one bend per edge.

On Embedding a Graph on Two Sets of Points

“…In this paper we continue the study, initiated in a previous work [1], of the apparently simple case of two colors. The two colors will be referred in the following as red and blue and the two set of points S 0 and S 1 will be denoted as R and B.…”

“…Therefore the question that we ask is whether O(n) bends per edge is also a lower bound for the 2CPSE problem and/or there are cases where a constant number of bends can be achieved. In [1] the simple family of bi-chromatic paths is considered and it is proved that every bi-chromatic path admits a 2CPSE on any two sets of red and blue points with at most one bend per edge.…”

k-Colored Point-Set Embeddability of Outerplanar Graphs

“…The intuition is that if the number of categories is constant, then a constant number of bends per edge may be sufficient, at least for simple classes of planar graphs. This type of investigation was started in [5,6], where the apparently simple case of k = 2 is studied. In [6] and in [5] a constant number of bends per edge is proved to be sufficient for constructing planar poly-line drawings of subclasses of outerplanar graphs, including paths, cycles, caterpillars, and wreaths.…”

k-colored Point-set Embeddability of Outerplanar Graphs

“…The intuition is that if the number of categories is constant, then a constant number of bends per edge may be sufficient, at least for simple classes of planar graphs. This type of investigation was started in [6,7], where the apparently simple case of k = 2 is studied. In [7] and in [6] a constant number of bends per edge is proved to be sufficient for constructing planar poly-line drawings of subclasses of outerplanar graphs, including paths, cycles, caterpillars, and wreaths.…”

“…This type of investigation was started in [6,7], where the apparently simple case of k = 2 is studied. In [7] and in [6] a constant number of bends per edge is proved to be sufficient for constructing planar poly-line drawings of subclasses of outerplanar graphs, including paths, cycles, caterpillars, and wreaths. In [6] it is also shown that there exists a 2-outerplanar graph G with a vertex partition V 0 , V 1 and two disjoint sets S 0 , S 1 of points such that any planar drawing of G that maps a vertex v ∈ V i to a distinct point of S i (i = 0, 1) has at least one edge with a linear number of bends.…”