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Planar Graphs

Tucker, Applied Combinatorics, Section 1.4, prepared by Patti Bodkin. Planar Graphs. a. This is a plane graph of Figure 1. a. b. e. e. c. f. b. d. d. f. c. Figure 1. A graph is called planar if it can be drawn on a plane without edges crossing.

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Planar Graphs

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  1. Tucker, Applied Combinatorics, Section 1.4, prepared by Patti Bodkin Planar Graphs Tucker, Applied Combinatorics, Sect. 1.4

  2. a This is a plane graph of Figure 1. a b e e c f b d d f c Figure 1 A graph is called planar if it can be drawn on a plane without edges crossing. A plane graph is a drawing of planar graph in the plane. Tucker, Applied Combinatorics, Sect. 1.4

  3. Here are two ways to determine if a graph is planar: The first is a practical heuristic, which the author refers to as the circle-chord method. It consists of a step-by-step method of drawing the graph, edge-by-edge without crossing any edges. The second consists of theoretical results, such as Kuratowski’s Theorem, or Euler’s Formula. Tucker, Applied Combinatorics, Sect. 1.4

  4. a b c d f e h g Circle-Chord Method--Planar Step One: Find a circuit that contains all the vertices of the graph. (Recall: a circuit is a path that ends where it starts) **Note: step one is not always possible, and is often difficult** The circuit for this figure is highlighted in red Tucker, Applied Combinatorics, Sect. 1.4 a a

  5. a e f a b c d b c h g d f e h g Circle-Chord Method--Planar Step Two: Draw this circuit as a large circle. Tucker, Applied Combinatorics, Sect. 1.4

  6. a e f a b c d b c h g d f e h g Circle-Chord Method--Planar Step Three: Choose one chord, and decide to draw it either inside or outside the circle. If chosen correctly, it will force certain other chords to be drawn opposite to the circle. (Inside if the first chord was drawn outside, and vice versa.) Since the chords could be drawn without crossing, this graph is planar. Tucker, Applied Combinatorics, Sect. 1.4

  7. a d b h c e f g Circle-Chord Method--Nonplanar Step One: Find a circuit that contains all the vertices of the graph. (Recall: a circuit is a path that ends where it starts) **Note: step one is not always possible, and is often difficult** The circuit for Figure 2 is highlighted in red Figure 2 Tucker, Applied Combinatorics, Sect. 1.4

  8. a a h d b b h c e c g f d f g e Circle-Chord Method--Nonplanar Step Two: Draw this circuit as a large circle. The remaining edges (in black) must be drawn either inside or outside the circle. These edges are called the chords. Tucker, Applied Combinatorics, Sect. 1.4

  9. a d b h c e f g Circle-Chord Method--Nonplanar Step Three: Choose one chord, and decide to draw it either inside or outside the circle. If chosen correctly, it will force certain other chords to be drawn opposite to the circle. (Inside if the first chord was drawn outside, and vice versa.) a Because these lines cross, this is not the plane graph for Figure 2. However, this does not mean that the graph is not planar. Often, it is very difficult to find the correct planar graph. h b c g d f e Tucker, Applied Combinatorics, Sect. 1.4

  10. Recall: and configurations. We say that a subgraph is a configuration if it can be obtained from a by adding vertices in the middle of some edges. A configuration is defined similarly. Here a has been subdivided by adding vertices in the middle of some of the edges. Tucker, Applied Combinatorics, Sect. 1.4

  11. Left Side Right Side Kuratowski’s Theorem A graph is planar if and only if it does not contain a subgraph that is a or configuration. The configuration is more common in nonplanar graphs than . If the graph is nonplanar than it contains one of the configurations with added vertices. a a b h b H and F were removed to show the configuration c c g g d d f e e Tucker, Applied Combinatorics, Sect. 1.4

  12. a b e d c A graph is defined to be a connected planar graph if it’s in “one-piece”, i.e. there are paths between every pair of vertices. Euler’s Formula Assume that is a connected planar graph, with:V = # of vertices E = # of edges R = # of regions. Euler’s Formula states that if is a connected planar graph, then any plane graph depiction of has regions. Figure 3: Tucker, Applied Combinatorics, Sect. 1.4

  13. a a a b b c c c e e a b e d c Euler’s Formula d Tucker, Applied Combinatorics, Sect. 1.4

  14. a c Then , , , and so Euler’s formula is valid for , since We obtain from by adding an edge at one of the vertices in . Proof of Euler’s Formula Draw a plane graph depiction of edge-by-edge. Let denote the (connected) plane graph obtained after n edges have been added, and let , , , denote the number of vertices, edges and regions in respectively. Initially there is , which consists of one edge, it’s two end vertices and the one (unbounded) region. Tucker, Applied Combinatorics, Sect. 1.4

  15. In general, is obtained from , by adding an edge at one of the vertices of . The new edge might link two vertices already in . If it does not, the other end vertex of the edge is a new vertex that must be added to . Now we will use the method of induction to complete the proof. We have shown that the theorem is true for . Next we assume that it is true for for any , and prove it is true for . Let be the edge that is added to to get . There are two cases to consider… Proof of Euler’s Formula (cont.) Tucker, Applied Combinatorics, Sect. 1.4

  16. In the first case, and are both in . Then they are on the boundary of a common region of , possibly the unbounded region. See Figure 4Edge splits into two regions. So each side of Euler’s formula grows by 1. Hence if the formula was true for , it will also be true for . Proof of Euler’s Formula (cont.) Then, Figure 4 Tucker, Applied Combinatorics, Sect. 1.4

  17. In the second case, one of the vertices is not in say it is . Then adding implies that is also added, but no new regions are formed (no existing regions are split). See Figure 5 Figure 5 Proof of Euler’s Formula (cont.) Thus and the value on each side of Euler’s formula is unchanged. The validity of Euler’s formula for implies its validity for . By induction, the formula is true for all and for the full graph . Tucker, Applied Combinatorics, Sect. 1.4

  18. Worked Example Book Example # 5 How many regions would there be in a plane graph with 10 vertices each of degree 3? (Recall: the degree of a vertex is the number of edges incident to the vertex) By the theorem in Section 1.3, the sum of the degrees, 10 x 3, equals , and so = 15. By Euler’s formula, the number of regions is: Tucker, Applied Combinatorics, Sect. 1.4

  19. Figure 6 Corollary to Euler’s Formula (Thm. 2) If is a connected planar graph with , then . Proof: The degree of a region is defined to be the number of edges on its boundary. To be precise, we say that if an edge occurs twice along a boundary, as does in region in Figure 6, the edge is counted twice in region degree; for example, region has degree 10 and region has degree 3 in the figure. Tucker, Applied Combinatorics, Sect. 1.4

  20. Each region in a plane graph must have degree , for a region of degree 2 would be bounded by 2 edges joining the same pair of vertices and a region of degree 1 would be bounded by a loop edge, but parallel edges and loops are not allowed in graphs (in this book.) Since each region has degree , the sum of degrees of all regions will be at least . But this sum of degrees of all regions must equal , since this sum counts each edge twice. That is each of an edge’s two sides is part of some boundary. Thus, = (sum of regions’ degrees) Tucker, Applied Combinatorics, Sect. 1.4

  21. Corollary to Euler’s Formula (Thm. 2) Proof: (cont.) Combining this inequality with Euler’s formula we have: Figure 4: Solving for e, we obtain: In this case, edge is counted twice in the deg( ). The deg( ) is 3. Tucker, Applied Combinatorics, Sect. 1.4

  22. CAREFUL! Do not interpret the corollary as meaning: If , then a connected graph is planar, because there are many nonplanar graphs which also satisfy this equation. For example, has 6 vertices and 9 edges. So when you substitute into the equation, you get: , which holds. However, is not planar. Tucker, Applied Combinatorics, Sect. 1.4

  23. However, the corollary states that must be true in a connected planar graph, and so the configuration cannot be planar. FOR THE CLASS TO TRY: Use the Corollary to Euler’s Formula to prove whether the following configuration is planar. Substituting into the Corollary: Configuration Tucker, Applied Combinatorics, Sect. 1.4

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