A curvature notion for planar graphs stable under planar duality

Testing Mutual Duality of Planar Graphs

We introduce and study the problem MUTUAL PLANAR DUALITY, which asks for two planar graphs G1 and G2 whether G1 can be embedded such that its dual is isomorphic to G2. Our algorithmic main result is an NP-completeness proof for the general case and a linear-time algorithm for biconnected graphs. To shed light onto the combinatorial structure of the duals of a planar graph, we consider the commo...

Treewidth of planar graphs: connection with duality

A graph is said to be chordal if each cycle with at least four vertices has a chord, that is an edge between two non-consecutive vertices of the cycle. Given an arbitrary graph G = (V, E), a triangulation of G is a chordal graph H(= V, F) such that E ⊆ F. We say that H is a minimal triangulation of G if no proper subgraph of H is a triangulation of G. The treewidth tw(H) of a chordal graph is i...

Treewidth of planar graphs: connections with duality

Planar Graph Characterization - Using γ - Stable Graphs

A graph G is said to be γ stable if γ(Gxy) = γ(G), for all x, y ∈ V (G), x is not adjacent to y, where Gxy denotes the graph obtained by merging the vertices x, y. In this paper we have provided a necessary and sufficient condition for Ḡ to be γ stable, where Ḡ denotes the complement of G. We have obtained a characterization of planar graphs when G and Ḡ are γ stable graphs. Key–Words: γ stable...

Weak Solutions for the Mean Curvature Flow of Planar Graphs