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 G be a planar graph such that |R| vertices of G are red and |B| vertices of G are blue. A bichromatic point-set embedding of G on R ∪ B is a crossing-free drawing of G such that each blue vertex is mapped to a point of B, each red vertex is mapped to a point of R, and each edge is a polygonal curve. We study the curve complexity of bichromatic point-set embeddings; i.e., the number of bends per edge that are necessary and sufficient to compute such drawings. We show that O(n) bends are sometimes necessary. We also prove that two bends per edge suffice if G is a 2-colored caterpillar and that for properly 2-colored caterpillars, properly 2-colored wreaths, 2-colored paths, and 2-colored cycles the number of bends per edge can be reduced to one, which is worst-case optimal.

This paper addresses the problem of designing drawing algorithms that receive as input a planar graph G, a partitioning of the vertices of G into k different semantic categories V0, · · · , V k−1 , and k disjoint sets S0, · · · , S k−1 of points in the plane with |Vi| = |Si| (i ∈ {0, · · · , k − 1}). The desired output is a planar drawing such that the vertices of Vi are mapped onto the points of Si and such that the curve complexity of the edges (i.e. the number of bends along each edge) is kept small. Particular attention is devoted to outerplanar graphs, for which lower and upper bounds on the number of bends in the drawings are established.

Abstract. This paper addresses the problem of designing drawing algorithms that receive as input a planar graph G, a partitioning of the vertices of G into k different semantic categories V0, · · · , V k−1 , and k disjoint sets S0, · · · , S k−1 of points in the plane with |Vi| = |Si| (i ∈ {0, · · · , k − 1}). The desired output is a planar drawing such that the vertices of Vi are mapped onto the points of Si and such that the curve complexity of the edges (i.e. the number of bends along each edge) is kept small. Particular attention is devoted to outerplanar graphs, for which lower and upper bounds on the number of bends in the drawings are established.

Hamiltonicity, book embeddability, and point-set embeddability of planar graphs are strictly related concepts. We exploit the interplay between these notions to describe colored sets of points and to design polynomial-time algorithms to embed kcolored planar graphs on these sets such that the resulting drawings have O(k) bends per edge.

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.