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1 6. Planarity Let G(V, E) be a graph with V = {v 1, v 2,..., v n } and E = {e 1, e 2,..., e m }. Let S be any surface (like the plane, sphere) and P = {p 1, p 2,..., p n } be a set of n distinct points of S, p i corresponding to v i, 1 i n. If e i = v j v k, draw a Jordan arc J i on S from p j to p k such that J i does not pass through any other p i. Then P {J 1, J 2,..., J m } is called a drawing of G on S, or a diagram representing G on S. The p i are called the points of the diagram and J i, the lines of the diagram. An embedding of a graph G on a surface S is a diagram of G drawn on the surface such that the Jordan arcs representing any two edges of G do not intersect except at a point representing a vertex of G. A graph is planar if it has an embedding on the plane. A graph which has no embedding on the plane is nonplanar. That is, a graph G is said to be planar if there exists some geometric representation of G which can be drawn on a plane such that no two of its edges intersect and a graph that cannot be drawn on a plane without a crossover between its edges is called nonplanar. In order that a graph G is nonplanar, we have to show that of all possible geometric representations of G, none can be embedded in a plane. Equivalently, a geometric graph G is planar if there exists a graph isomorphic to G that is embedded in a plane. An embedding of a planar graph G on a plane is called a plane representation of G. Figure 6.1 shows three diagrams of the same graph which is planar. The two graphs in Figure 6.2 represent the same planar graph. Fig. 6.1 Fig. 6.2

2 Graph Theory Kuratowski s Two Graphs The complete graph K 5 and the complete bipartite graph K 3,3 are called Kuratowski s graphs, after the polish mathematician Kasimir Kurtatowski, who found that K 5 and K 3,3 are nonplanar. Theorem 6.1 The complete graph K 5 with five vertices is nonplanar. Proof Let the five vertices in the complete graph be named v 1, v 2, v 3, v 4, v 5. Since in a complete graph every vertex is joined to every other vertex by means of an edge, there is a cycle v 1 v 2 v 3 v 4 v 5 v 1 that is a pentagon. This pentagon divides the plane of the paper into two regions, one inside and the other outside, Figure 6.3(a). Since vertex v 1 is to be connected to v 3 by means of an edge, this edge may be drawn inside or outside the pentagon (without intersecting the five edges drawn previously). Suppose we choose to draw the line from v 1 to v 3 inside the pentagon, Figure 6.3(b). In case we choose outside, we end with the same argument. Now we have to draw an edge from v 2 to v 4 and another from v 2 to v 5. Since neither of these edges can be drawn inside the pentagon without crossing over the edge already drawn, we draw both these edges outside the pentagon, Figure 6.3(c). The edge connecting v 3 and v 5 cannot be drawn outside the pentagon without crossing the edge between v 2 and v 4. Therefore v 3 and v 5 have to be connected with an edge inside the pentagon, Figure 6.3(d). Now, we have to draw an edge between v 1 and v 4 and this cannot be placed inside or outside the pentagon without a crossover. Thus the graph cannot be embedded in a plane. Fig. 6.3

3 136 Planarity Theorem 6.2 The complete bipartite graph K 3,3 is nonplanar. Proof The complete bipartite graph has six vertices and nine edges. Let the vertices be u 1, u 2, u 3, v 1, v 2, v 3. We have edges from every u i to each v i, 1 i 3. First we take the edges from u 1 to each v 1, v 2 and v 3. Then we take the edges between u 2 to each v 1, v 2 and v 3, Figure 6.4(a). Thus we get three regions namely I, II and III. Finally we have to draw the edges between u 3 to each v 1, v 2 and v 3. We can draw the edge between u 3 and v 3 inside the region II without any crossover, Figure 6.4(b). But the edges between u 3 and v 1, and u 3 and v 2 drawn in any region have a crossover with the previous edges. Thus the graph cannot be embedded in a plane. Hence K 3,3 is nonplanar. Fig. 6.4 We observe that the two graphs K 5 and K 3,3 have the following common properties. 1. Both are regular. 2. Both are nonplanar. 3. Removal of one edge or a vertex makes each a planar graph. 4. K 5 is a nonplanar graph with the smallest number of vertices, and K 3,3 is the nonplanar graph with smallest number of edges. Thus both are the simplest nonplanar graphs. The following result given independently by Fary [77] and Wagner [260] implies that there is no need to bend edges in drawing a planar graph to avoid edge intersections. Theorem 6.3 (Fary) Every triangulated planar graph has a straight line representation. Proof The proof is by induction on the number of vertices. The result is obvious for n = 4. So, let n 5 and assume that the result is true for all planar graphs with fewer than n vertices. Let G be a plane graph with n vertices. First, we show that G has an edge e belonging to just two triangles. For this, let x be any vertex in the interior of a triangle T and choose x and T such that the number of regions inside T is minimal.

4 Graph Theory 137 Let y be a neighbour of x, and the edge xy lies inside T, and let xy belong to three triangles xyz 1, xyz 2 and xyz 3. Then one of these triangles lies completely inside another. Assume that z 3 lies inside xyz 1. Then z 3 and xyz 1 contradict the choice of x and T (Fig. 6.5). Fig. 6.5 Thus there is an edge e = xy lying in just two triangles xyz 1 and xyz 2. Contracting xy to a vertex u, we get a new graph G with a pair of double edges between u and z 1, and u and z 2. Remove one each of this pair of double edges to get a graph G which is a triangulated graph with n 1 vertices. By the induction hypothesis, it has a straight line representation H. The edges of G correspond to uz 1, uz 2 in H. Divide the angle around u into two parts in one of which the pre-images of the edges adjacent to x in G lie, and in the other, the pre-images of the edges adjacent to y in G. Thus u can be pulled apart to x and y, and the edge xy is restored by a straight line to get a straight line representation of G. 6.2 Region A plane representation of a graph divides the plane into regions (also called windows, faces or meshes). A region is characterised by the set of edges (or the set of vertices) forming its boundary. We note that a region is not defined in a nonplanar graph, or even in

5 138 Planarity a planar graph not embedded in a plane. Thus a region is a property of the specific plane representation of a graph and not of an abstract graph. Fig. 6.6 Infinite region: The portion of the plane lying outside a graph embedded in a plane such as region 4 in the graph given in Figure 6.6, is infinite in its extent. Such a region is called an infinite, unbounded, outer or exterior region for that particular plane representation. Like other regions, the infinite region is also characterised by a set of edges (or vertices). Clearly, by changing the embedding of a given planar graph, we can change the infinite region. Consider the graphs of Figure 6.7. In the two embeddings of the same graph, finite region v 1 v 3 v 5 in (a) is infinite in (b). Fig. 6.7 Embedding on a sphere: A graph is spherical if it can be embedded on the surface of a sphere. The following result implies that a planar graph can often be embedded on the surface of a sphere, so that there is no distinction between finite and infinite region. Theorem 6.4 in a plane. A graph can be embedded on the surface of a sphere if it can be embedded Proof Consider the stereographic projection of a sphere on the plane. Put the sphere on the plane and call the point of contact as SP (south-pole). At point SP, draw a straight line perpendicular to the plane, and let the point where this line intersects the surface of the sphere be called NP (north-pole).

6 Graph Theory 139 Fig. 6.8 Now, corresponding to any point p on the plane, there exists a unique point p on the sphere, and vice versa, where p is the point where the straight line from point p to point NP intersects the surface of the sphere. Thus there is a one-one correspondence between the points of the sphere and the finite points on the plane, and points at infinity in the plane corresponding to the point NP on the sphere. Therefore from this construction, it is clear that any graph that can be embedded in a plane can also be embedded on the surface of the sphere, and vice versa. We now have the following result. Theorem 6.5 A planar graph can be embedded in a plane such that any specified region, specified by the edges forming it, can be made the infinite region. Proof A planar graph embedded on the surface of the sphere divides the surface of the sphere into different regions. Each region of the sphere is finite, the infinite region on the plane having been mapped onto the region containing the point NP. Clearly, by suitably rotating the sphere, we can make any specified region map onto the infinite region on the plane. Hence the result. Remark Thinking in terms of the regions on the sphere, we observe that there is no real difference between the infinite region and the finite regions on the plane. Therefore, when we discuss the regions in a plane representation of a graph, we include the infinite region. Also, since there is no essential difference between an embedding of a planar graph on a plane, or on a sphere (a plane can be regarded as the surface of the sphere of infinitely large radius), the term plane representation of a graph is often used to include spherical as well as plane embedding. 6.3 Euler s Theorem The following important result due to Euler gives a relation between the number of vertices, edges, regions and the components of a planar graph. Theorem 6.6 then If G is a planar graph with n vertices, m edges, f regions and k components, n m+ f = k+1. (6.6.1)

7 140 Planarity Proof We construct the graph G by the addition of successive edges starting from the null graph K n. For this starting graph, k = n, m = 0, f = 1, so that (6.6.1) is true. Let G i 1 be the graph at the start of ith stage and G i be the graph obtained from G i 1 by addition of the ith edge e. If e connects two components of G i 1, then f is not altered, m is increased by 1 and k is reduced by 1, so that (6.6.1) holds for G i as it holds for G i 1. If e joins two vertices of the same components of G i 1, k is unaltered, m is increased by 1 and f is increased by 1, so that again (6.6.1) holds for G i. The following relation between the number of vertices, edges and regions is the discovery of Euler [75] and is also called as Euler s formula for planar graphs. Theorem 6.7 (Euler) In a connected planar graph with n vertices, m edges and f regions (faces), n m+ f = 2. (The proof can also be deduced from Theorem 6.6 by taking k = 1, as the connected graph has one component.) Proof Without loss of generality, assume that the planar graph is simple. Since any simple planar graph can have a plane representation such that each edge is a straight line, any planar graph can be drawn such that each region is a polygon (a polygon net). Let the polygon net representing the given graph consist of f regions. Let k p be the number of p-sided regions. Since each edge is on the boundary of exactly two regions, 3k 3 + 4k 4 + 5k rk r = 2m, (6.7.1) where k r is the number of polygons with r edges. Also, k 3 + k 4 + k k r = f. (6.7.2) The sum of all angles subtended at each vertex in the polygon net is 2πn. (6.7.3.) Now, the sum of all interior angles of a p-sided polygon is π(p 2) and the sum of the exterior angles is π(p+2). The expression in (6.7.3) is the total sum of all interior angles of f 1 finite regions plus the sum of the exterior angles of the polygon defining the infinite region. This sum is π(3 2)k 3 + π(4 2)k π(r 2)k r + 4π = π[3k 3 + 4k rk r + 2(k 3 + k k r )]+ 4π = π(2m 2 f)+4π = 2π(m f + 2). (6.7.4) Equating (6.7.3) and (6.7.4) we get 2π(m f + 2) = 2nπ, so that f = m n 2. Definition: Let φ be a region of a planar graph G. We define the degree of φ, denoted by d(φ), as the number of edges on the boundary of φ.

8 Graph Theory 141 Second proof of Theorem 6.7 We use induction on m, the number of edges. If m = 0, then G is K 1, a graph with 1 vertex and 1 region. So, n m+ f = = 2, and the result is true. If m = 1, then the number of vertices in G is either one or two, the first possibility occurring when the edge is a loop. These two possibilities give rise respectively to two regions and one region. Fig. 6.9 { 1 1+2, in the loop case, Therefore, n m+ f = 2 1+1, in the non loop case. Thus, n m + f = 2, and again the result is true. Assume that the result is true for all connected planar graphs with fewer than m edges. Let G have m edges. Case I Suppose G is a tree. Then m = n 1 and f = 1, because a planar representation of a tree has only one region. Thus, n m+ f = n (n 1)+1 = 2, and the result holds. Case II Suppose G is not a tree. Then G has cycles. Let C be a cycle in G. Let e be an edge of C. The graph G e has one edge less than the graph G. Also the number of vertices in G e and G are same. Since removing e coalesces two regions in G into one in G e, therefore G e has one region less than that in G. Thus by induction hypothesis in G e, we have n (m 1)+(f 1) = 2, so that n m+ f = 2. Hence the result. The next result gives an upper bound for the edges of a simple planar graph. Theorem 6.8 Let G be a simple planar graph with n vertices and m edges, where n 3. Then m 3n 6. Proof First assume that the planar graph G is connected. If n = 3, then since G is simple, therefore G has at most three edges. Thus m 3, that is, m 3 3 6, and the result is true. Now, let n = 4. First let G be a tree so that m = n 1. Since n 4, obviously we have n 1 3n 6. Now, let G be not a tree. Then G has cycles. Clearly, there is a cycle in G, all of whose edges lie on the boundary of the exterior region of G. Then, since G is simple, we have d(φ) 3, for each region φ of G. Let b = d (φ), where F denotes the set of all regions of G. φ F

9 142 Planarity Since each region has at least three edges on its boundary, therefore we have b 3 f, where f is the number of regions of G. However, when we sum to get b, each edge of G is counted either once or twice (twice when it occurs as a boundary edge for two regions). Therefore, b 2m so that 3 f b 2m. In particular, we have 3 f 2m. Using Euler s formula n m+ f = 2, we get m 3n 6. Let G be not connected and let G 1, G 2,..., G t be its connected components. For each i, 1 i t, let n i and m i denote the number of vertices and edges in G i. Since each G i is a planar simple graph, by the above argument, we have m i 3n i 6, for each i, 1 i t. Also, n = t n i and m = t m i. i=1 i=1 Hence, m = t m i t (3n i 6) = 3 t n i 6t 3n 6. i=1 i=1 i=1 The following result gives the existence of a vertex of degree less than six in a simple planar graph. Theorem 6.9 If G is a simple planar graph, then G has a vertex v of degree less than 6. Proof If G has only one vertex, then this vertex has degree zero. If G has only two vertices, then both vertices have degree at most one. Let n 3. Assume degree of every vertex in G is at least six. Then, We know d (v) 6n. v V(G) d (v) 2m. v V(G) Thus, 2m 6n so that m 3n. This is not possible because, by Theorem 6.8, we have m 3n 6. contradiction. Hence G has at least one vertex of degree less than 6. Thus we get a Corollary 6.1 K 5 is nonplanar. Proof Here n = 5 and m = 10. So, 3n 6 = 15 6 = 9. Thus m > 3n 6. Therefore K 5 is nonplanar. Corollary 6.2 K 3, 3 is nonplanar. Proof Since K 3, 3 is bipartite, it contains no odd cycles, and so no cycle of length three. It follows that every region of a plane drawing of K 3, 3 if it exists, has at least four boundary edges. We have d(φ) 4, for each region φ of G.

10 Graph Theory 143 Let b = d (φ), where F denotes the set of all regions of G. Since each region has at φ F least four edges on its boundary, we have b 4 f, where f is the number of regions of G. Now, when we sum up to get b, each edge of G is counted either once or twice, and so b 2m. Thus, 4 f b 2m, so that 2f m. For K 3,3, we have m = 9, and so 2 f 9 giving f 9. But by Euler s formula, f = 2 m n+2 = = 5, a contradiction. Hence K 3,3 is nonplanar. Euler s formula gives a necessary condition for connected planar graphs. Theorem 6.10 For a connected simple planar graph with n vertices and m edges, and g(n 2) girth g, we have m g 2. Proof Let G be a connected simple planar graph with n vertices, m edges and girth g. Let F be the set of regions and d(φ) be the degree of the region φ. Then d (φ) = 2m. Let φ F f be the number of regions, that is F = f. As g is the length of the smallest cycle in G, therefore g is the smallest degree of a region. So, d (φ) g f. Thus, 2m g f. Now, using φ F g(n 2) Euler s formula, f = m n+2, we have 2m g(m n+2), so that m g 2. Corollary 6.3 The Peterson graph P is nonplanar. Proof The girth ( of the ) Perterson graph P is 5, n(p) = 10, and m(p) = 15. Thus, if P were 10 2 planar, 15 = 5, which is not true. Hence P is nonplanar. 5 2 Corollary 6.4 Any graph containing a homeomorph of a K 5 or K 3, 3 is nonplanar. This follows from the fact that any homeomorph of a nonplanar graph is nonplanar. The following result characterises nonplanar 3-connected graphs with minimum edges. Theorem 6.11 A nonplanar connected graph G with minimum number of edges that contains no subdivision of K 5 or K 3, 3 is simple and 3-connected. Proof Since G has minimum number of edges, G is minimal with respect to these properties. The minimality of G ensures that it is simple. Also, a graph is planar if and only if each of its blocks is planar. This implies that a minimal nonplanar graph is a block. Suppose such a graph has a two vertex cut S = {u, v}. Then G S is disconnected. Let G 1 be one of its components and G 2 be the union of all of its other components. Let H 1 = < V(G 1 ) S > + e and H 2 = < V(G 2 ) S > + e, where e = uv. Clearly, these are nonempty graphs with fewer edges than G. Since G is nonplanar, at least one of H 1 and H 2 is nonplanar. If both are planar, let H 1 be a plane embedding of H 1 and let φ be a region of H 1 containing the edge e.

11 144 Planarity A plane embedding H 2 of H 2 inside φ can then be obtained with the edge e of both H 1 and H 2 coinciding. Then H 1 H 2 e is a plane embedding of G, contradicting our hypothesis. Suppose H 1 is nonplanar. Since m(h 1 ) < m(g), therefore H 1 contains a subdivision K of either K 5 or K 3, 3 according to the hypothesis. Clearly e E, since otherwise K G, contradicting the assumption. Now, replacing e of K by a u v path in H 2, we get a homeomorph of K contained in G. This is again a contradiction and shows that G has no 2-vertex cut. Hence G is 3-connected. The following result is used in proving Kuratowski s theorem. Theorem 6.12 a. If G e contains a subdivision of K 5, then G contains a subdivision of K 5 or K 3, 3. b. If G e contains a subdivision of K 3, 3, then G contains a subdivision of K 3, 3. Proof Let G = G e be a graph obtained by contracting the edge e = xy of G. Let w be the vertex of G obtained by contracting e = xy. a. Let G e contains a subdivision of K 5, say H. If w is not a branch vertex of H, then G also contains a subdivision of K 5, obtained by expanding w back into the edge xy, if necessary (Fig. 6.10). Fig Assume w is a branch vertex of H and each of x, y is incident in G to two of the four edges incident to w in H. Let u 1 and u 2 be the branch vertices of H that are at the other ends of the paths leaving w on edges incident to x in G. Let v 1, v 2 be the branch vertices of H that are at the other ends of the paths leaving w on edges incident to y in G (Fig. 6.11). By deleting the u 1 u 2 path and v 1 v 2 path from H, we obtain a subdivision of K 3, 3 in G, in which y, u 1, u 2 are branch vertices for one partite set, and x, v 1, v 2 are branch vertices of the other.

12 Graph Theory 145 Fig b. Let G e contain a subdivision of K 3, 3, say H. If w is not a branch vertex of H, then G also contains a subdivision of K 3, 3, obtained by expanding w back into the edge xy, if necessary (Fig. 6.12). Fig Now, assume that w is a branch vertex in H and at most one of the edges incident to w in H is incident to x in G. Then w can be expanded into xy to lengthen that path and y becomes the corresponding branch vertex of K 3, 3 in G (Fig. 6.13). Fig Kuratowski s Theorem This theorem was independently given by Kuratowski [144] and Frink and Smith. In 1954, Dirac and Schuster [69] found a proof that was slightly shorter than the original proof. The proof given here is due to Thomassen [241].

13 146 Planarity Theorem 6.13 (Kuratowski) subdivision of K 5 or K 3, 3. A graph is planar if and only if it does not have any Proof Necessity Let G be a planar graph. Then any of its subgraphs is neither K 5 nor K 3, 3 nor does it contain any subdivision of K 5 or K 3, 3. Sufficiency It is enough to prove sufficiency for 3-connected graphs. Let G be a 3- connected graph with n vertices. We prove that the 3-connected graph G either contains a subdivision of K 5 or K 3, 3 or has a convex plane representation. This we prove by using induction on n. Since G is 3-connected, therefore n 4. For n = 4, G = K 4 and clearly has a plane representation. Now, let n 5. Assume the result to be true for all 3-connected graphs with fewer than n vertices. Since G is 3-connected, G has an edge e such that G e is 3-connected. Let e = xy. If G e contains a subdivision of K 5 or K 3, 3, then G also contains a subdivision of K 5 or K 3, 3. Therefore, let G e = H have a convex plane representation. Let z be the vertex obtained by contraction of e = xy. The plane graph obtained by deleting the edges incident to z has a region containing z (this may be the exterior region). Let C be the cycle of H z bounding this region. Since we started with a convex plane representation of H, we have straight segments from z to all its neighbours. Let x 1, x 2,..., x k be the neighbours of x in that order on C. If all the neighbours of y belong to a single segment from x i to x i+1 on C, then we obtain a convex plane representation of G by putting x at z in H, and putting y at a point close to z in the wedge formed by x x i and x x i+1. If all the neighbours of y do not belong to any single segment x i x i+1 on C (1 i k, x k+1 = x 1 ), then we have the following cases (Fig. 6.14). a. y shares three neighbours with x. In this case C together with these six edges involving x and y form a subdivision of K 5. b. y has two u, v in C that are in different components of the subgraph of C obtained by deleting x i and x i+1, for some i. In this case, C together with the paths uyv, x i xx i+1 and xy form a subdivision of K 3, 3. Fig. 6.14

14 Graph Theory Geometric Dual Let G be a plane graph. The dual of G is defined to be the graph G constructed as follows. To each region f of G there is a corresponding vertex f of G and to each edge e of G there is corresponding edge e in G such that if the edge e occurs on the boundary of the two regions f and g, then the edge e joins the corresponding vertices f and g in G. If the edge e is a bridge, i.e., the edge e lies entirely in one region f, then the corresponding edge e is a loop incident with the vertex f in G. For example, consider the graph shown in Figure Fig Theorem 6.14 The dual G of a plane graph is planar. Proof Let G be a plane graph and let G be the dual of G. The following construction of G shows that G is planar. Place each vertex fk of G inside its corresponding region f i. If the edge e i lies on the boundary of two regions f j and f k of G, join the two vertices f j and fk by the edge e i, drawing so that it crosses the edge e exactly once and crosses no other edge of G (Fig. 6.16). Fig Remarks Clearly, there is one-one correspondence between the edges of plane graph G and its dual G with one edge of G intersecting one edge of G.

15 148 Planarity 1. An edge forming a self-loop in G gives a pendant edge in G (An edge incident on a pendant vertex is called a pendant edge). 2. A pendant edge in G gives a self loop in G. 3. Edges that are in series in G produce parallel edges in G. 4. Parallel edges in G produce edges in series in G. 5. The number of edges forming the boundary of a region f i in G is equal to the degree of the corresponding vertex f i in G. 6. Considering the process of drawing a dual G from G, it is evident that G is a dual of G. Therefore, instead of calling G a dual of G, we usually say that G and G are dual graphs. 7. Let n, m, f, r and µ denote the number of vertices, edges, regions, rank and nullity of a connected plane graph G and let n, m, f, r and µ be the corresponding numbers in G. Then n = f, m = m, f = n. We have r = n 1, µ = m n + 1, r = n 1, µ = m n+1. So, r = f 1, µ = m f + 1, r = n 1, µ = m n+1. Using Euler s formula, n m + f = 2 or f = m n + 2, we have r = m n 2 1 = m n+1 = µ and µ = m f + 1 = n+ f 2 f + 1 = n 1 = r. We now have the following result. Theorem 6.15 The edge e is a loop in G if and only if e is a bridge in G. Proof Let the edge e be a loop in a plane graph G. Then it is the edge on the common boundary of two regions on which, say f, lies within the area of the plane surrounded by e with the other, say g, lying outside the area. Thus, from definition of G, e is the only path from f to g in G. Thus e is a bridge in G. Conversely, let e be a bridge in G, joining vertices f and g. Thus e is the only path in G from f to g. This implies, again from the definition of G, that the edge e in G completely encloses one of the regions f and g. So e is a loop in G. Remark The occurrence of parallel edges in G is easily described. Given two regions f and g of G, there are k parallel edges between f and g if and only if f and g have k edges on their common boundary. Note We have defined the dual of a plane graph instead of a planar graph. The reason for this is that different plane drawings G 1 and G 2 of the same planar graph G may result in non-isomorphic duals G 1 and G 2. 2-isomorphism Two graphs G(V, E)and H(W, F) are said to be 2-isomorphic if there exists a bijection φ: E F such that both φ and φ 1 preserve cycles.

16 Graph Theory 149 Two graphs G and H are said to be 2-isomorphic if there exists a one-one correspondence between their edge sets such that the edges of a cycle in G correspond to the edges of a cycle in G 2 and vice versa. Example Figure 6.17(a) shows 2-isomorphic graphs. 1-isomorphism Graphs which become isomorphic after the splitting of all of their cut vertices are said to be 1-isomorphic. Example Figure 6.17(b) shows 1-isomorphic graphs. Fig Consider the two non-isomorphic graphs G 1 and G 2 shown in Fig. 6.17(a) and (b). They both have a single cut vertex, namely v. If this cut vertex is split into two vertices in each of G 1 and G 2, we obtain the edge disjoint graphs G 1 and G 2 as shown in (c) and (d). Clearly, G 1 and G 2 are isomorphic. Thus G 1 and G 2 are 1-isomorphic. The next result gives a correspondence between the edges of a graph with the edges of its dual. Theorem 6.16 Edges in a plane graph G form a cycle in G if and only if the corresponding dual edges form a bond in G.

17 150 Planarity Proof Let D E(G). It is sufficient to prove that D contains a cycle if and only if the set D of dual edges contains a bond of G. Now, let D contain a cycle C. Then by the Jordan Curve theorem, some region of G lies inside C and some region lies outside C. These regions correspond to vertices v, w in G, one drawn inside C and one outside C. A v w path in G crosses C and hence uses an edge of G, that is dual to an edge of C. Thus D disconnects v from w and hence D contains a bond (Fig. 6.18(a)). Conversely, let D contain a bond. We show that D contains a cycle. Assume that D does not contain a cycle. Then D encloses no region. It remains possible to reach each region of G from every other without crossing D. Hence G D is connected, and so D contains no bond. This is a contradiction. Hence D contains a cycle. Fig. 6.18(a) Theorem 6.17 All duals of a planar graph G are 2-isomorphic and any graph H 2-isomorphic to a dual G of G is itself a dual of G. Proof Let G (V, E ) be a dual of G. Then there is an edge bijection φ : E E such that cycles in G correspond to bonds in G. Let H(W, F) be any other dual of G. Then there is an edge bijection ψ: E E such that bonds in G correspond to cycles in H. Therefore φ ψ : E F is an edge bijection which preserves cycles between G and H. Hence G and H are 2-isomorphic. Conversely, let H be a graph which is 2-isomorphic to a dual G of G. Then there is an edge bijection χ : E F which preserves cycles between G and H. Also, there is an edge bijection φ : E E such that cycles in G correspond to bonds in G. Thus we have an edge bijection χφ : F E such that cycles in F correspond to bonds in G. Hence H is a dual of G. The following result provides the condition for two given graphs to be duals of each other. Theorem 6.18 A necessary and sufficient condition for two planar graphs G 1 and G 2 to be the duals of each other is that there should be a one-one correspondence of the edges in G 1 with the edges in G 2 such that the subset of edges in G 1 forms a cycle if and only if the corresponding set of edges in G 2 forms a bond. Proof Necessity Consider a plane representation of a planar graph G 1 and let G 1 be the dual of G 1. Let C be an arbitrary cycle in G 1.

18 Graph Theory 151 Clearly, C will form some closed simple curve in the plane representation of G 1, dividing the plane into two areas. Thus the vertices of G 1 are partitioned into two non-empty, mutually exclusive subsets, one inside C and the other outside. In other words, the set of edges C in G 1 corresponding to the set C in G 1 is a bond in G 1. Similarly, corresponding to a bond S in G 1, there is a unique cycle consisting of the corresponding edge set S in G 1 such that S is a cycle. Sufficiency Let G be a planar graph, and let G be a graph for which there is a one-one correspondence between the bonds of G and the cycles of G and vice versa. Let G be the dual graph of G. Therefore there is a one-one correspondence between the bonds of G and cycles of G. So there is a one-one correspondence between the cycles of G and G. Thus G and G are 2-isomorphic. Hence G is the dual of G. Let G be a plane graph and G be the dual of G. So to every vertex of G we have a region of G. Let the region φ of the plane graph G corresponding to the vertex v of G has e 1, e 2,..., e k as its boundary edges. Then each of these edges e i crosses the corresponding edge e i of G, and these edges are all incident at the vertex v. It follows that φ contains the vertex v. As G is a plane graph, we can construct the dual of G, called the double dual of G and is denoted by G. We now have the following observation. Theorem 6.19 dual G. Let G be a plane connected graph. Then G is isomorphic to its double Proof Let G be a plane connected graph and G be the dual of G. Now, any region φ of dual G contains exactly one vertex of G, namely its corresponding vertex v of G. This is because the number of the regions of G is same as the number of vertices of G. Thus in the construction of G, we take the vertex v to be the vertex in G corresponding to the region φ of G. This choice gives the required isomorphism. Definition: A connected graph G is called self-dual if it is isomorphic to its dual G. For example, K 4 is self-dual. The next result is a characterisation of bipartite graphs in terms of duality. Theorem 6.20 Eulerian. A connected plane graph G is bipartite if and only if its dual graph G is Proof Let G be a bipartite graph. Then G does not contain odd cycles. Since G is a plane, all the regions of G are of even length. So degree of every region in G is even. Let G be the dual graph of G. Since the vertex degrees of G are same as the corresponding degrees of the regions of G, degree of every vertex in G is even. Hence G is Eulerian. Conversely, assume G is Eulerian. Then degree of every vertex in G is even. Since the degree of a vertex in G corresponds to the degree of the region in G, degree of every region in G is even. This shows that G contains only even cycles. Hence G is bipartite.

19 152 Planarity Dual of a subgraph: Let G be a plane graph and let G be its dual. Let e be an edge in G and let e be the corresponding edge in G. We delete the edge e and find dual of G e. If edge e is on the boundary of two regions, then removal of e merges these two regions into one. Thus the dual (G e) is obtained from G by deleting the corresponding edge e and then fusing the two end vertices of e in G e. On the other hand, if edge e is not on the boundary, then e forms a self-loop. In that case, G e is the same as (G e). Thus, if a graph G has a dual G, then the dual of any subgraph of G can be obtained by successive application of this procedure. Dual of a homeomorphic graph: Let G be a plane graph and G be its dual. Let e be an edge in G and let e be the corresponding edge in G. We create an additional vertex in G by introducing a vertex of degree two in edge e (e now becomes two edges in series). This simply adds an edge parallel to e in G. Likewise, the reverse process of merging two edges in series will simply eliminate one of the corresponding parallel edges in G. Thus, by this procedure, if the graph G has dual G, the dual of any graph homeomorphic to G can be obtained from G. These are illustrated in Figure 6.18(b). The following important result is due to Whitney [267]. Theorem 6.21 (Whitney) A graph has a dual if and only if it is planar. Proof We have only to prove that a nonplanar graph does not have a dual. Let G be a nonplanar graph. Then according to Kuratowski s theorem, G contains K 5 or K 3, 3, or a sub-division of K 5 or K 3, 3. We know that a graph G has a dual only if every subgraph H of G and every subdivision of G has a dual. Thus to prove the result, we show that neither K 5 nor K 3, 3 has a dual. Fig. 6.18(b)

20 Graph Theory 153 a. Suppose that K 3, 3 has a dual D. Then the bonds in K 3, 3 correspond to cycles in D and vice-versa. Since K 3, 3 has no bond consisting of two edges, D has no cycle consisting of two edges. That is, D contains no pair of parallel edges. Since every cycle in K 3, 3 is of length four or six, D has no bond with less than four edges. Therefore the degree of every vertex in D is at least four. As D has no parallel edges and the degree of every vertex is at least four, D has at least five vertices each of degree four or more ( 4). That is, D has at least = 10 edges. This is a contradiction to the fact that K 3, 3 has nine edges. Hence K 3, 3 has no dual. b. Suppose K 5 has a dual H. We note that K 5 has i. 10 edges, ii. no pair of parallel edges, iii. no bond with two edges, and iv. bonds with only four or six edges. Therefore graph H has i. 10 edges, ii. no vertex with degree less than three, iii. no bond with two edges, and iv. cycles of length four or six only. Now graph H contains a hexagon (a cycle of length six) and no more than three edges can be added to a hexagon without creating a cycle of length three, or a pair of parallel edges. Since both of these are not present in H, and H has 10 edges, there must be at least seven vertices in H. The degree of each of these vertices is at least three. This implies that H has at least 11 edges, which is a contradiction. Hence K 5 has no dual. Definition: Two plane graphs G and G are said to be duals (or combinatorial duals) of each other if there is a one-one correspondence between the edges of G and G such that if H is any subgraph of G, and H is the corresponding subgraphs of G, then rank (G H ) = rank G nullity H. 6.6 Polyhedron A polyhedron is a solid bounded by surfaces, called faces, each of which is a plane. That is, a solid bounded by plane surfaces is called a polyhedron. For example, brick, window frame, tetrahedron. A polyhedron is said to be convex if any two of its interior points can be joined by a straight line lying entirely within the interior. For example, a brick is convex, but a window frame is not convex.

21 154 Planarity The vertices and edges of a polyhedron, which form a skeleton of the solid, give a simple graph in three dimensional space. It can be shown that for a convex polyhedron this graph is planar. To see this, imagine the faces of the convex polyhedron to be made of rubber, with one face, say at the base, missing. Then taking hold of the edges at the missing face, we are able to stretch out the rubber to form one plane sheet. The vertices and edges in this transformation now form a plane graph. Moreover, each face of this plane graph corresponds to a face of a solid, with the exterior face of the graph corresponding to the missing face we used in stretching process. Clearly, the plane graph is also connected, the degree of every vertex is at least 3 and the degree of every face is also at least 3. Also, the graph is simple. A simple connected plane graph G is called polyhedral if d(v) 3, for each vertex v of G, and d(φ) 3, for every face φ of G. The following is a simple but useful property of polyhedral graphs. Theorem 6.22 Let P be a convex polyhedron and G be its polyhedral graph. For each n 3, let n i denote the number of vertices of G of degree i and let f i denote the number of faces of G of degree i. Then a. in i = i f i = 2m, where m is the number of edges of G. i 3 i 3 b. The polyhedron P, and so the graph G, has at least one face bounded by a cycle of length n for either n = 3, 4 or 5. Proof a. The expression in i is simply i 3 We know, v V(G) d(v) = 2m, so that in i = 2m. v V (G) i 3 d(v), since d(v) 3 for each vertex v. For each n 3, the expression i f i is obtained by going round the boundary of each face having a cycle of length i as its boundary, counting up the edges as we go. If we do this over all possible i, then we count each edge twice, and so i f i = 2m. i 3 b. Assume to the contrary that P has no faces bounded by cycles of length 3, 4 or 5. Then f 3 = f 4 = f 5 = 0. So by (a), 2m = i f i 6 f i = 6 f i = 6 f, where f is the total number of faces i 6 i 6 i 6 of P. Thus, f 1 3 m. Also by (a), 2m = in i 3n i = 3 n i = 3n, where n is the total number of i 3 i 3 i 3 vertices of P. Thus, n 2 3 m. Now, by Euler s formula, m = n+ f 2 and we have

22 Graph Theory 155 m 2 3 m+ 1 m 2 = m 2, and this implies that 2 0, 3 which is impossible. Hence P must have a face of length 3, 4 or 5. Regular polyhedra: A polyhedron is called regular if it is convex and its faces are congruent regular polygons (so that the polyhedral angles are all equal). The regular polyhedra are also called Platonic bodies or Platonic solids. A fact that has been known for at least 2000 years is that there are only five regular polyhedra, which is proved next by using graph theoretic argument. Theorem 6.23 The only regular polyhedra are the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. Proof Let P be a regular polyhedron and let G be its corresponding polyhedral graph. Let n, m and f denote the number of vertices, edges and faces of G. Since the faces of G are congruent to each other, each one is bounded by the same number of edges. Thus, d(φ) = k, k 3 for each face φ of G. Also, we know that there is at least one cycle of length 3, 4 or 5, therefore, 3 k 5. Similarly, since the polyhedral angles are all equal to each other, the graph G is regular, say r-regular, where r 3. Thus we have rn = 2m and k f = 2m, so that 2m = rn = k f. (6.23.1) Using this in Euler s formula n m+ f = 2, we have 8 = 4n 4m+4f = 4n 2m+4f 2m = 4n rn+4f k f. Therefore, 8 = (4 r)n+(4 k) f. (6.23.2) Since n and f are both positive and also 3 k 5 and r = 3, there are only five possible cases (Fig. 6.19). Case 1 Let r = 3 and k = 3. Then 8 = (4 3)n+(4 3) f, so that 8 = n+ f. From ( ), n = f. So, n+ f = 8 gives n = 4, f = 4. Thus, n = 4, m = 6, f = 4, which is a tetrahedron. Case 2 Let r = 3, k = 4. Then from (6.23.2), 8 = (4 3)n+(4 4) f. So, n = 8. Then, (6.23.1) gives 3n = 4 f, so that f = 6. Also, m = 12. So, n = 8, m = 12, f = 6, and thus is a cube.

23 156 Planarity Fig. 6.19

24 Graph Theory 157 Case 3 Let r = 3, k = 5. Then (6.23.2) gives 8 = (4 3)n+(4 5) f, so that 8 = n f. From (6.23.1), 3n = 5 f. Solving 8 = n f and 3n = 5 f, we get n = 20, f = 12. Thus, n = 20, m = 30, f = 12, which is a dodecahedron. Case 4 Let r = 4, k = 3. Then we get n = 6, f = 8. So, n = 6, m = 12, f = 8, which is an octahedron. Case 5 Let r = 5, k = 3. Then we have n = 12, f = 20. Thus, n = 12, m = 30, f = 20, which is an icosahedron. The following is an elegant necessary condition for a plane graph to be Hamiltonian and is due to Grinberg [86]. Theorem 6.24 If G is a loopless plane graph having a Hamiltonian cycle C, then n (i 2)(φ i φ i ) = 0, where φ i and φ i are the number of regions of G of degree i contained in interior C and exterior C respectively. Proof Let E and E denote the sets of edges of G contained in int Cand ext C, respectively, and let E = m and E = m. Then int C contains exactly m +1 regions (Fig. 6.20) and so n i=2 φ i = m + 1, (6.24.1) Since G is loopless, φ 1 = φ 1 = 0. i=2 Fig. 6.20

25 158 Planarity Also, each edge in int C is on the boundary of exactly two regions in int C and each edge of C is on the boundary of exactly one region in int C. Thus, counting the edges of all the regions in int C, we obtain n iφ i = 2m + n. (6.24.2) i=2 Eliminating m between (6.24.1) and (6.24.2), we get n i=2 (i 2)φ i = n 2. (6.24.3) Similarly, n i=2 (i 2)φ i = n 2. (6.24.4) The required result follows from equation (6.24.3) and (6.24.4). Example Consider the Herschel graph G shown in Figure Clearly, G has 9 regions and all the regions are of degree 4. Thus, if G were Hamiltonian, then 2(φ 4 φ 4 ) = 0. This implies φ 4 = φ 4, which is impossible, since φ 4 + φ 4 = number of regions of degree 4 in G = 9 is odd. Hence G is non-hamiltonian. Fig Konig observed that every graph is embeddable on some orientable surface. This is seen by drawing an arbitrary graph G on the plane, possibly with edges that cross over each other and then attaching a handle to the plane at each crossing and allowing one edge to go over the handle and the other under it. Konig further showed that any embedding of a graph on an orientable surface with a minimum number of handles has all its regions simply connected. Obviously, planar graphs can be embedded on a sphere. A toroidal graph is a graph embedded on a torus, for instance, K 5, K 3, 3, K 7 and K 4, 4.

26 Graph Theory 159 Definition: The genus γ(g)of a graph G is the minimum number of handles which are added to the sphere so that G can be embedded on the resulting surface. Obviously γ(g) = 0 if and only if G is planar. The homeomorphic graphs have the same genus. The genus of a polyhedron is the number of handles required on the sphere for a surface to contain the polyhedron. Definition: The least number of planar subgraphs whose union is a given graph G is called the thickness of G. Clearly the thickness of any planar graph is 1. Also, K 5 and K 3, 3 have thickness 2, while the thickness of K 9 is 3. The crossing number of a graph G is the minimum number of pairwise intersections of its edges when G is drawn in the plane. Clearly, crossing number of a graph is zero if and only if G is planar. The following result is due Euler and the proof can be found in Courant and Robbins [61]. Theorem 6.25 For a polyhedron of genus γ with n vertices, m edges and f regions n m+ f = 2 2γ. We have the following observations. If G is a connected graph of genus γ, then m = 3(n 2+2γ), i f every region in G is a triangle, 2(n 2+2γ), i f every region in G is a quadrilateral. From this, we have the following. If G is a connected graph of genus γ, then γ 1 6 m 1 2 (n 2), and if G has no triangles, then γ 4 1m 1 2 (n 2). The proof of the following result is due to Ringel and Youngs [223] and some details can be found in Harary [104]. Theorem 6.26 For n 3, the genus of the complete graph is γ(k n ) = (n 3)(n 4) Decomposition of Some Planar Graphs Schnyder [232] proved that each triangulated planar graph G can be decomposed into three edge disjoint trees. If instead triangles all faces (or regions) of G are quadrilaterals, and G is without loops and multiple edges, Petrovic [183] proved that two trees are sufficient.

27 160 Planarity The following two lemmas are used in proving Theorem Lemma 6.1 If G is a planar graph on n(n 4) vertices whose all regions are quadrilaterals, then E(G) = 2n 4. Lemma 6.2 If G is a planar graph on n(n 4) vertices whose all regions are quadrilaterals, then G contains at least three vertices of degree 3. The following result is due to Petrovic [183]. Theorem 6.27 (Petrovic) Let G be a planar graph on n(n 4) vertices whose all faces are quadrilaterals, and v r and v b any two non-adjacent vertices of G (we assume v r is coloured red and v b blue). Then the edges of G can be partitioned into red and blue ones so that red ones form a spanning tree T r of G v b, and blue ones a spanning tree T b of G v r. Proof We call T r the red, and T b the blue tree. A vertex of G is red (respectively blue) if it belongs to T r (respectively T b ). A vertex is red-blue if it belongs to both T r and T b. Thus v r is the only red, and v b is only blue vertex in G. We proceed by induction on n. For n = 4, the assertion is obvious. Assume that it holds for each graph on less than n(n > 4) vertices, and consider a graph G on n vertices. Let u (u v r, v b ) be a vertex of G of degree 3 existing by Lemma 6.2. a. d(u) = 2. Let uv 1, uv 2 be the edges, and uv 1 xv 2 u, uv 1 yv 2 u the faces incident with u. Since G C 4, each of vertices v 1 and v 2 has degree 3. It implies that all regions of G u are quadrilaterals. By induction hypothesis G u can be partitioned into trees T r and T b. Now, independent of the kind of vertices v 1 and v 2, (red, blue or red-blue) we can always colour the edges uv 1 and uv 2 so that the obtained trees T r and T b are as desired. b. d(u) = 3. Let uv 1, uv 2, uv 3 be the edges and uv 1 xv 2 u, uv 2 yv 3 u, uv 3 zv 1 u be the faces incident with u. First assume that all vertices x, y, z are distinct. Since no triangle can be partitioned into disjoint quadrilaterals whose vertices are vertices of the triangle and some its inner points, xy, yz, zx / E(G). Suppose one of vertices x, y, z, is blue and there is no edge connecting it with the opposite vertex of the hexagon v 1 xv 2 yv 3 z. Without loss of generality, we may assume it is x. Then xv 3 / E(G). Denote by G the graph obtained from G by deleting the edges xv 1 and xv 2 and identifying vertices u and x. Since each face of G is a quadrilateral it can be partitioned into two trees T r and T b, where V(T r )= V(G v b ) and V(T b ) = V(G v r ). If edges uv 1 and uv 2 are coloured differently in G, say uv 1 red and uv 2 blue, we colour xv 1 red and xv 2 blue obtaining the required trees T r = T r + xv 1 and T b = T b +xv 2 in G. Therefore assume that uv 1 and uv 2 are monocoloured, say red. Then one of vertices v 1 and v 2, say v 1, is red blue. As u is also a red-blue vertex, there is the unique blue u v 1 path P in G. If P contains uv 3, we colour xv 1 and xv 2 blue and red, respectively. If not, we colour both edges xv 1 and xv 2 red, and recolour uv 1 blue. It is routine to check that obtained trees T r = T r +xv 1, T b = T b +xv 2 in the first case, and T r = T r uv 1 +xv 1 +xv 2, T b = T b + uv 1 in the second, are as required.

28 Graph Theory 161 Now, assume none of the vertices x, y, z satisfy the condition above. Since at most one of edges xv 3, yv 1, zv 2 can exist, we may assume without loss of generality that x is red-blue, y is the red, z is the blue vertex, xv 3 E(G) and yv 1 / E(G). Then each region of the graph G obtained from G by deleting edges yv 2 and yv 3 and identifying y and u is a quadrilateral. By induction ypothesis, G can be composed into trees T r and T b, where V(T ) = V(G z) and V(T b ) = V(G y). Colouring all edges incident with y red, and recolouring the edge uv 3 blue, we obtain the decomposition of G into trees T r and T b, where V(T r ) = V(G z) and V(T b ) = V(G y). Suppose two of vertices x, y, z, say y and z, coincide. Then d(v 3 ) = 2. If v 3 is redblue, we finish the proof as in case (a). If x is red-blue we proceed as above. So we may assume that {v 3, x} = {v r, v b ). Let v 3 = v r, x = v b. By induction hypothesis the graph G IV = G v 3 can be decomposed into trees Tr IV and Tb IV, where V(T IV r ) = V(G IV x) and V(Tb IV ) = V(GIV u). Then edges uv 1, uv 2 are coloured red and edges xv 1, xv 2, blue. It implies that edges yv 1, yv 2 are coloured differently, yv 1, red and yv 2, blue. Now, colouring v 3 u and v 3 y red and recolouring uv 1 blue, we get from Tr IV and Tb IV the desired trees T r and T b, where V(T r ) = V(T r ) = V(G x) and V(T b ) = V(G v 3 ). If x = y = z, then v(g) = {u, x, v 1, v 2, v 3 }, E(G) = {uv 1, uv 2, uv 3, xv 1, xv 2, xv 3 } and the statement holds trivially. 6.8 Exercises 1. If G is a connected planar graph of order less than 12, prove that δ(g) = If G is a planar graph of order 24 and is regular of degree 3, then what is the number of regions in a planar representation of G? 3. Prove that Euler s formula fails for disconnected graphs. 4. Show that every graph with at most three cycles is planar. 5. Find a simple graph G with degree sequence [4, 4, 3, 3, 3, 3] such that a. G is planar, b. G is nonplanar. 6. Show that every simple bipartite cubic planar graph contains a C Prove that a simple planar graph has at least 4 vertices of degree 5 at most. 8. Let G be a planar, triangle free graph of order n. Prove that G has no more than 2n 4 edges. 9. Let G be of order n = 11. Show that at least one of G or G is non planar. 10. Show that the average degree of a planar graph is less than Use Kuratowski s theorem to prove that the Peterson graph is nonplanar.

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