Discrete trace theorems and energy minimizing spring embeddings of planar graphs

Geodesic Embeddings and Planar Graphs

Geodesi Embeddings and Planar Graphs

Chordal embeddings of planar graphs

Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result.

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...

Maintaining alterable planar embeddings of dynamic graphs

We present a data structure for maintaining a planar embedding of a dynamic plane graph. The graph may be updated with edge deletions and insertions of edges when it does not violate the embedding. Queries of vertex pairs return the list of possible places for inserting an edge. Furthermore, the embedding may be updated with flips that alter the embedding by swapping the orientation of a subgra...