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🔗. 2.3 Planar Graphs. 🔗. Objectives. After completing this section, you should be able to do the following. Distinguish between planar and non-planar graphs. Use Euler’s formula to prove that certain graphs are non-planar. Apply Euler’s formula to polyhedra. 🔗. Section Preview. 🔗. Investigate!
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- Properties of Planar Graphs
- Non-planar Graph
- Properties of non-planar Graphs
- Graph Coloring
- Applications of Graph Coloring
- State and Prove Handshaking Theorem.
If a connected planar graph G has e edges and r regions, then r ≤e.If a connected planar graph G has e edges, v vertices, and r regions, then v-e+r=2.If a connected planar graph G has e edges and v vertices, then 3v-e≥6.A complete graph Knis a planar if and only if n<5.A graph is said to be non planar if it cannot be drawn in a plane so that no edge cross. Example:The graphs shown in fig are non planar graphs. These graphs cannot be drawn in a plane so that no edges cross hence they are non-planar graphs.
A graph is non-planar if and only if it contains a subgraph homeomorphic to K5 or K3,3 Example1: Show that K5is non-planar. Solution: The complete graph K5contains 5 vertices and 10 edges. Now, for a connected planar graph 3v-e≥6. Hence, for K5, we have 3 x 5-10=5 (which does not satisfy property 3 because it must be greater than or equal to 6). Th...
Suppose that G= (V,E) is a graph with no multiple edges. A vertex coloring of G is an assignment of colors to the vertices of G such that adjacent vertices have different colors. A graph G is M-Colorable if there exists a coloring of G which uses M-Colors. Proper Coloring:A coloring is proper if any two adjacent vertices u and v have different colo...
Some applications of graph coloring include: 1. Register Allocation 2. Map Coloring 3. Bipartite Graph Checking 4. Mobile Radio Frequency Assignment 5. Making a time table, etc.
Handshaking Theorem:The sum of degrees of all the vertices in a graph G is equal to twice the number of edges in the graph. Mathematically it can be stated as: ∑v∈Vdeg(v)=2e Proof: Let G = (V, E) be a graph where V = {v1,v2, . . . . . . . . . .} be the set of vertices and E = {e1,e2. . . . . . . . . .} be the set of edges. We know that every edge l...
Draw, if possible, two different planar graphs with the same number of vertices and edges, but a different number of faces. When is it possible to draw a graph so that none of the edges cross? If this is possible, we say the graph is planar (since you can draw it on the plane).
When a connected graph can be drawn without any edges crossing, it is called planar. When a planar graph is drawn in this way, it divides the plane into regions called faces. Draw, if possible, two different planar graphs with the same number of vertices, edges, and faces.
Apr 11, 2022 · A planar graph is one that can be drawn in a plane without any edges crossing. For example, the complete graph K₄ is planar, as shown by the “planar embedding” below.
- Russell Lim
A key property of planar graphs is that they can be embedded in the plane in such a way that no edges overlap. Planar graphs can have at most 3V - 6 edges for any planar graph with V vertices, where V ≥ 3. Not all graphs are planar; for example, K5 and K3,3 are known to be non-planar graphs.
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Figure 1.2: Planar, non-planar and dual graphs. (a) Plane ‘butterfly’graph. (b, c) Non-planar graphs. (d) The two red graphs are both dual to the blue graph but they are not isomorphic. Image source: wiki. Given a graph G,itsline graph or derivative L[G] is a graph such that (i) each vertex
