Universal Sets for Straight-Line Embeddings of Bicolored Graphs

Upward Straight-Line Embeddings of Directed Graphs into Point Sets

In this paper we consider the problem of characterizing the directed graphs that admit an upward straight-line embedding into every point set in convex or in general position. In particular, we show that no biconnected directed graph admits an upward straight-line embedding into every point set in convex position, and we provide a characterization of the Hamiltonian directed graphs that admit u...

Straight Line Embeddings of Planar Graphs on Point Sets

Labeling Subgraph Embeddings and Cordiality of Graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

Graph Drawings with Relative Edge Length Specifications

We study plane straight-line embeddings of graphs where certain edges are specified to be longer than other edges. We analyze which graphs are universal in the sense that they allow a plane embedding for any total, strict order on the edge lengths. In addition, we also briefly consider circular arc drawings with relative edge length specifications.

On Universal Point Sets for Planar Graphs

A set P of points in R is n-universal, if every planar graph on n vertices admits a plane straight-line embedding on P. Answering a question by Kobourov, we show that there is no n-universal point set of size n, for any n 15. Conversely, we use a computer program to show that there exist universal point sets for all n 10 and to enumerate all corresponding order types. Finally, we describe a col...