Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a planar map.

Jan 17, 2025 · Planar graphs and graph coloring are key concepts in graph theory with diverse applications in fields like computer science and engineering, focusing on the properties of graphs that can be drawn without edge crossings and the assignment of colors to vertices to avoid adjacent similarities.

Planar Graphs and Planar Maps (1) Observation A planar graph has a dual graph which is also planar. In the dual graph, the concepts of faces and vertices are interchanged.

Base case: A map with just one area, produces a planar graph. Inductive Hypothesis: Assume that any map with N areas produces a planar graph. Inductive Step: Uses inductive hypothesis to prove that any graph with N+1 area will also produce a planar graph.

Planar graphs originated with the studies of polytopes and of maps. The skeleton (edges) of a three-dimensional polytope provide a planar graph. We obtain a planar graph from a map by representing countries by vertices, and placing edges between countries that touch each other. Assuming each country is contiguous, this gives a planar graph.

Fitting planar graphs on planar maps is related to cluster planarity [3, 4, 14]. In a cluster-planar drawing, (also called c-planar drawing in the literature), a plane graph (i.e., a planar graph with a xed planar embedding) along with a clustering (i.e., a partition of the vertices) is given.

We consider a problem that combines aspects of both by studying the problem of fitting planar graphs on planar maps. After providing an NP-hardness result for the general decision problem, we identify sufficient conditions so that a fit is possible.

Planar maps = Def. Planar map = connectedmultigraphembedded on the sphere A map is easier to draw in the plane (implicit choice of anouter face f 0)) (up to continuous deformation) 6= Rk: a planar graph can have several embeddings on the sphere f 0 f 0 a map has vertices, edges, andfaces 5 faces (including outer one)

graphs: are they “planar”? A planar graph is a graph which can be drawn in the plane without any edges crossing? For example, K4 is planar, cube (Q3) is planar, but K3,3 isn’t. Notice that some pictures of a planar graph may have crossing edges. What makes it planar is that you can draw at least one picture of the graph with no crossings.

Planar graphs and planar maps. A planar embedding of a graph \(G\) represents its vertices as distinct points in the plane (typically drawn as small circles) and its edges as simple interior-disjoint paths between their endpoints.