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A 1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and a k-planar graph is a graph that may be drawn with at most k simple crossings per edge. A map graph is a graph formed from a set of finitely many simply-connected interior-disjoint regions in the plane by connecting two regions when they ...
The concept of a combinatorial map was introduced informally by J. Edmonds for polyhedral surfaces [2] which are planar graphs.It was given its first definite formal expression under the name "Constellations" by A. Jacques [3] [4] but the concept was already extensively used under the name "rotation" by Gerhard Ringel [5] and J.W.T. Youngs in their famous solution of the Heawood map-coloring ...
Let H = {h 1, h 2, ..., h k} be the convex hull of P; then the farthest-point Voronoi diagram is a subdivision of the plane into k cells, one for each point in H, with the property that a point q lies in the cell corresponding to a site h i if and only if d(q, h i) > d(q, p j) for each p j ∈ S with h i ≠ p j, where d(p, q) is the Euclidean ...
For disconnected graphs, definitions vary: the diameter may be defined as infinite, or as the largest diameter of a connected component, or it may be undefined. diamond The diamond graph is an undirected graph with four vertices and five edges. diconnected Strong ly connected. (Not to be confused with disconnected) digon
A 1-planar drawing of the Heawood graph: six of the edges have a single crossing, and the remaining 15 edges are not crossed.. In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge.
In graph theory, an st-planar graph is a bipolar orientation of a plane graph for which both the source and the sink of the orientation are on the outer face of the graph. . That is, it is a directed graph drawn without crossings in the plane, in such a way that there are no directed cycles in the graph, exactly one graph vertex has no incoming edges, exactly one graph vertex has no outgoing ...
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. [9] Such a drawing is called a plane graph or planar embedding of the graph.
The Tutte polynomial factors into connected components. If is the union of disjoint graphs and ′ then = ′ If is planar and denotes its dual graph then (,) = (,)Especially, the chromatic polynomial of a planar graph is the flow polynomial of its dual.
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