This paper presents two special classes of plane graphs characterized by the sequences M(v) and W(e). M(v) is the circular sequence consisting of the numbers of vertices on the meshes around vertex v ; and W(e) is the sequence consisting of the numbers of vertices on the meshes to the right and the left of edge e and also of the degrees of the head and the tail of e. A graph is called uniform with respect to M(v) or W(e) if its vertices all have the same M(v), or if its edges all have the same W(e), respectively. It is shown that if such a uniform plane graph exists for the given M(v) or W(e), the numbers of its vertices, edges and meshes are uniquely determined. Then, the conditions on M(v) or W(e) for the existence of a graph are investigated. Tables of plane graphs which are uniform with respect to M(v) or W(e) are presented. Besides regular polyhedrons, there are thirteen types of graphs which are uniform with respect to M(v), and only four graphs which are uniform with respect to W(e)

AKAIKE, Shuji

OZAWA, Takao

English

Kyoto University Research Information Repository

RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Uniform Plane GraphsOZAWA, Takao; AKAIKE, ShujiOZAWA, Takao ...[et al]. Uniform Plane Graphs. Memoirs of the Facultyof Engineering, Kyoto University 1978, 39(4): 495-5031978-01-31http://hdl.handle.net/2433/281051Uniform Plane Graphs By Takao OZA WA* and Shuji AKAIKE** (Received June 27, 1977) Abstract This paper presents two special classes of plane graphs characterized by the sequences M(v) and W(e). M(v) is the circular sequence consisting of the num-bers of vertices on the meshes around vertex v; and W(e) is the sequence con-sisting of the numbers of vertices on the meshes to the right and the left of edge e and also of the degrees of the head and the tail of e. A graph is called uniform with respect to M(v) or W(e) if its vertices all have the same M(v), or if its edges all have the same W (e), respectively. It is shown that if such a uniform plane graph exists for the given M(v) or W(e), the numbers of its vertices, edges and meshes are uniquely determined. Then, the conditions on M(v) or W(e) for the existence of a graph are investigated. Tables of plane graphs which are uniform with respect to M (v) or W (e) are presented. Besides regular polyhedrons, there are thirteen types of graphs which are uniform with respect to M(v), and only four graphs which are uniform with respect to W (e). 1. Introduction 495 A graph is often characterized by numbers associated with its vertices and edges. For instance, the degree of a vertex is the number of edges incident with the vertex, and a graph is called regular (vertex-regular) if all its vertices have the same de-gree.1> The sequence of numbers consisting of the degrees of the vertices is called the degree sequence, and the synthesis of a graph having the given degree sequence is often considered by graph theorists. Now a graph is planar if it can be embedded in the plane, and if it has already been embedded in the plane, it is called a plane graph.t> For a plane graph, numbers associated with its meshes can also be used to characterize the graph. In this paper, we consider special classes of plane graphs which are character-ized by constant sequences of numbers associated with vertices or edges. Let G be a plane graph. Although G is not directed, we treat each edge of G as two directed * Department of Electrical Engineering II. ** Department of Electrical Engineering II, (Present Address:· Toyoh Electronics Co. Ltd., Tokyo) 496 Takao OZAWA and Shuji AKAIKE parallel edges of opposite direction.2> If one of the directed edges is e, the other is denoted by e'. The sequences of numbers M(v) and W(e) defined below are 'associ-ated with a vertex v and a directed edge e in G, respectively. Definition: M(v) is the circular sequence consisting of the numbers of vertices on the meshes around v. The order of the numbers in M(v) corresponds to the order of the meshes drawn on the plane. Definition : W(e) is the sequence of four numbers which sequence consists of the numbers of vertices on the meshes to the right and the left of e, and of the degrees of the head and the tail of e. The numbers in W(e) are ordered as given above. Since M(v) is a circular sequence, its expression by a linear sequence is not unigue. For example, the expression of M(v 1 ) for vertex v, in graph G,, shown in Fig. 1, may be {3, 3, 5, 4}, {3, 5, 4, 3}, {5, 4, 3, 3}, or {4, 3, 3, 5}. These se-quences are regarded all equal and any one of them can represent M(v1). One way to recognize whether two circular sequences represent the same M(v) or not, is to move the numbers in the sequences cyclically so that they are in a certain lexicographic order. It may also happen that a number in M(v) has no one-to-one Fig. 1. Graph G 1 correspondence with a mesh around v. For example, M(v 2 ) for vertex v2 in G, is {4, 3, 4, 3}. The first number 4 in this sequence can be the number of vertices on either of the two meshes, m 1 and m 2 • As an example of W(e), W(e,)={5, 4, 3, 4} for edge e, in G,. If W(e) is given, W(e') can be determined. For example, W(e,') ={4, 5, 4, 3}. Graph G is called uniform with respect to M(v), if its vertices all have the same M(v). We treat each edge as two directed edges e and e'. Graph G is called uniform with respect to W(e), if its edges all have the same W(e) and W(e'). W(e) and W(e') may not be the same. 2. Uniform Graphs with respect to M(v) We are first interested in uniform plane graphs with respect to M(v). They are regular graphs. We give the definitions of the symbols as follows. G: uniform plane graph with respect to M(v) V: number of vertices in G E: number of edges in G F: number of meshes in G F;: number of meshes in G which have L; vertices R;: number of meshes around a vertex which have L; vertices D : degree of a vertex Then V-E+F=2 F="5:;. F1 I E=DV/2 D="2;.R1 I F1=R1V/L;. Uniform Plane Graphs ............ ( 1) ............ (2) ............ ( 3) ............ ( 4) ............ ( 5) 4r.Jl Eq. (1) is the famous Euler's theorem for polyhedrons. The derivation of eq. (5) is as follows. There are R 1 meshes around each vertex which have L; vertices, and there are V vertices in G. Since each mesh contributes L1 timei; in R1 V, the total number of meshes which have L; vertices is R; VIL;. The rest of the equations are easily obtained from the definitions. In order to avoid any unnecessary complication we assume D23 and L123, which means that there are no series edges or parallel edges in G. Eliminating E and F from eqs. (1) - (5), we get v- 2 1-D/2+ "5:;,(R;/L;) ............ ( 6) i If M(v) is given, D, R;, and L; can be determined. Thus eq. (6) shows that V can be determined uniquely for the given M(v). Then E and F can also be deter-mined from eqs. (3) and (1). Since V must be a positive interger, there are certain restrictions on M(v). First 1 >D/2-"5:;, (R;/ L;) >-D!2-"5:;,R1! 3=D/6. .. .......... (7) i i Therefore D can be 3, 4 or 5 only. Especially if there is no triangular mesh in G, then L124 and we obtain l>D/4 instead of inequality (7). Thus D must be 3 only. Further conditions on M(v) and the graphs for M(v) sati~fying the conditions are investigated below. Case a. All the meshes around vertex v have the same number of vertices: In this case G represents a regular polyhedron, and we get Table 1. Case b. At least one of the meshes around vertex v has a number of vertices different from those of the other meshes. In the following, A, B and C represent all different numbers. We have: Proposition 1. Suppose vertices v, and v, are adjacent, as shown in Fig. 2. M(v,) contains subsequence {A, B}, and M(v,) contains subsequence {B, A}. G is uniform and M(v,)=M(v2 ) if the numbers in the sequences are properly moved. Therefore M(v) must contain either {A, B, A}, {B, A, B} or {A, B, ... , B, A}. Case b-1. D=3. From Proposition 1 at least two numbers in M(v) must be the 498 Takao OZAWA and Shuji AKAIKE Table. 1. regular polyhedrons M(v) W(e) D V E F I G {3,3,3} {3,3,3,3} 3 4 6 4 I tetrahedron {4,4, 4} {4,4,3,3} I 3 8 12 I 6 I hexahedron {5, 5, 5} {5, 5, 3, 3} 3 20 30 12 dodecahedron {3,3,3,3} {3,3,4,4} 4 I 6 12 8 octahedron {3,3,3,3,3} I {3,3,5,5} I 5 I 12 I 30 20 icosahedron same. Let M(v) ={A. A, B}(={A, B, A}). Then from eq. (6) and V>O, V 2 2A -----~ and B<-~- • 2/A+l/B-1/2 A-4 ............ ( 8) Since B~3, 3s.As.11. (1) A=3; M(v) ={3, 3, B}: Suppose vertex v 1 has three adjacent vertices V2, V3 and v4 as shown in Fig. 3. We see that the degree of V2, v3 or v, is 3, and there can be no mesh other than those shown in the figure. Therefore B cannot be more than 3. (2) A=4; M(v) ={4, 4, B}: We obtain graphs as shown in Fig. 4. For each B~3, G is uniquely determined. (3) A=6; M(v) ={6, 6, B}: For each of B=3, 4 and 5, G is uniquely deter-mined. See Fig. 5, Fig. 6 and Fig. 7. (4) A=8; M(v)={8, 8, B}: G is uniquely determined for B=3. See Fig. 8. (5) We can check that no graph exists for A=5, 7 and 11. Case b--2. D=4. From Proposition 1 the types of M(v) are restricted to {A, A, A, B}, {A, A, B, B}, {A, B, A, B} and {A, B, A, C}. Case b-2-i. M(v) ={A, A, A, B}: From eq. (6) and V>O, Fig. 2. Graph for Propo-sition 1. V2-----Fig. 3. Graph for M(v)= U,3,B}. 8 Fig. 4. Graph for M(v)= {4,4, B} Uniform Plane Graphs 499 Fig. 5. Graph for M(v)={6,6,3}. Fig. 6. Graph for M(v)={6,6,4}. Fig. 7. Graph for M(v)={6,6,5}. Fig. 8. Graph for M(v)={8,8,3}. 2 A V 3/ A+l!B-1 and B< A-3 • . ........... ( 9 ) Since B~3. A=3 or 4. (1) A=3; M(v)={3, 3, 3, B}: G can be uniquely determined for each B~4. (2) A=4; M(v) ={4, 4, 4, B}: B can be 3 only. There are two non-isomorphic plane graphs for M(v) ={4, 4, 4, 3} as shown in Fig. 9. Case b-2-ii. M(v) ={A, A, B. B} and {A, B, A, B}: From eq. (6) and V>O. 2 2A V- 2/A+2!B-1 and B< A-.,f. (o) ( b) Fig. 9. Graph for M(v)={4,4,4,3}. . ........... (10) Fig. 10. Graph for M(v)={4, 3, 4, 3}, W(e)={3,4,4,4}. 500 Takao OZAWA and Shuji AKAIKE Since B-:?.3, A<6. (1) A=4, B=3; M(v)={4, 3, 4, 3}: G is uniquely determined. See Fig. 10. (2) A=5, B=3; M(v) ={5, 3, 5, 3}: G is uniquely determined. (3) No graph other than those given above can exist in this case. Case b-2-iii. M(v) ={A, B, A, C}: From eq. (6) and V>O, V 2 2/A+l(B+l/C-r and 2/A+l/B+l!C>I. ............ (11) G can exist for M(v) ={4, 5, 4, 3} only. For this M(v), G is unique. Case b-3. D=5: From Proposition 1, the possible types of M(v) are {A, A. A, A, B}, {A, A, A, B, B}, {A, A, B, A, B}, {A, A, B, A, C} and {A, B, B, A, C}. Case b-3-i. M(v) ={A, A, A, A, B}: From eq. (6) and V>O, 2 2A V= 4/A+l/B-3/2 and B< 3A-8 • ············U2) Since B?:3, A=3 only. B can be 4 or 5. It can be checked that a graph can really exist for each of M(v) ={3, 3, 3, 3, 4} and {3, 3, 3, 3, 5} and is uniquely determined. Case b-3-ii. From eq. (6) we see that V can not be positive for the rest of the types of M(v) under the condition that A, B, C2:3 and A'i'=Bci=C'i'=A. The results obtained for Case b are summarized in Table 2. Table 2. Uniform plane graphs with respect to M (v). M(v) I D I V I E I F I G {4,4,B} B=3,5,6,··· I 3 I 2B I 3B I B+2 I Fig. 4 {6, 6, 3} I 3 I 12 I 18 I 8 I Fig. 5 {6,6, 4} I 3 I 24 I 36 I 14 I Fig. 6 {6, 6, 5} I 3 I 60 I 90 I 32 I Fig. 7 {8,8,3} I 3 I 24 I 36 I 14 I Fig. 8 {10,10,3} I 3 I 60 I 90 I 32 I {3, 3, 3, B} B=4, 5, ··· I 4 I 2B I 4B 2B+2 {4,4,4,3} I 4 24 I 48 26 Fig. 9 {4,3,4,3} I 4 12 I 24 I 14 I Fig. 10 {5, 3, 5, 3} I 4 I 30 I 60 I 32 I {5, 4, 3, 4} I 4 60 I 120 I 62 I {1, 3, 3, 3, 4} I 5 24 I 60 I 48 I {3, 3, 3, 3, 5} I 5 I 60 I 150 I 92 I Uniform Plane Graphs 501 3. Uniform Graphs with respect to W(e) In this chapter, we consider uniform plane graphs with respect to W(e). Let G be such a graph. Eqs. (1) and (2) in the previous chapter also hold for G. In the following, A or B is the number of vertices in a mesh and A*B, and I or J is the degree of a vertex and I cf:.]. D is the degree of a vertex when G is regular. Case a. W(e) = W(e') ={A, A, D, D}: We obtain regular polyhedrons in this case. See Case a and Table 1 in the previous chapter. Case b. W(e) ={A, B, D, D} and W(e') ={B, A, D, D}: G is a regular graph, and eqs. (3) - (5) in the previous chapter also hold. We can assume A <B without loss of generality. The meshes with A vertices and those with B vertices appear alternately around a vertex, and therefore D must be even. Let D=2K. Since D?:.3, K?:_2. The numbers of meshes with A vertices and B vertices are both equal to K. From eqs. (3) and (5) we have E=KV, F.A=KV!A and Fn=KV/B. .. .......... (13) where FA and F n are numbers of meshes with A vertices and B vertices in G respec-tively. Then from eqs. (1). (2) and (13). V-~~--2~- ~ - . 1-K+K!A+K!B .. .......... (14) From eq. (14) we see that A must be 3 m order to get a positive V, and that B is either 4 or 5. (1) A=3, B=4; W(e) ={3, 4, 4, 4}: G is uniquely determined and is shown in Fig. 10. This graph has a uniform M(v) of {4, 3, 4, 3}. (2) A=3, B=5; W(e) ={3, 5, 4, 4}: G is unique and has a uniform M(v) of {5, 3, 5, 3}. Case c. W(e) ={A, A. I, J} and W(e') ={A, A, ], I}: Let Gd be the dual graph of G. Then of course G is the dual of Gd. Let ed be the edge in Gd corres-ponding to the edge e in G. If W(e) ={A, A, I, J}, then W(ed) in Gd is {I, ], A, A}. From this fact we see that the graphs which satisy the condition of Case c are the duals of those obtained in Case b. Thus we get graphs for W(e) ={4, 4, 3, 4,} and W(e) ={4, 4, 3, 5}, which are the duals of those for W(e) ={3, 4, 4, 4} and W(e) ={3, 5, 4, 4} respectively. The graph for W(e) ={4, 4, 3, 4} is shown in Fig. 11. Case d. W(e) = {A, B, I, J} and W(e') = {B, A, ], I}: Since there can be no edge which is incident with two vertices of the same degree, the graph satisfying the Fig. 11. Graph for W (e)={4, 4, 3, 4}. 502 Takao OZAWA and Shuji AKAIKE condition of this case, if any, must be a bipartite graph. The vertices of G are partitioned into two sets, namely V1 and VJ' , consisting of the vertices with degree I and J respectively. Each vertex in V1 is adjacent to I vertices in VJ', and each vertex in V1 to J vertices in VJ'. The numbers of vertices in V1 and VJ' must be JV I (I+ f) and IV! (I+ f) respectively. The total number of edges of G is, then, E= IJV! (I+ f). The numbers of meshes with A vertices around a vertex in V1 and VJ' are l/2 and J/2 respectively, and then the total number of meshes with A vertices is IJV/2 (I+ f)A+ JIV/2 (I+ J)A=E/A. Similarly, the total number of meshes with B vertices is E! B. Therefore, F=E! A+ E! B and it follows from eq. (I) that V 2 Since Az:3, Bz:3 and Act=B, it is required ............ (16) in order for V>O. On the other hand, since meshes with A vertices and those with B vertices appear alternately around a vertex, I and J must be even numbers, and Iz:4 and Jz:6, or Iz:6 and Jz:4. Thus IJ!Cl+ ])";?12/5. We see that inequality (16) cannot be satisfied and there is no graph for Case d. The results obtained for Case b and Case c are summarized in Table 3. Table 3. Uniform plane graphs with respect to W(e). W(e) V I E I F I G {3,4,4, 4} 12 I 24 14 I Fig.JO {3,5,4,4} I 30 I 60 32 I {4,4,3,4} 14 I 24 I 12 I Fig.11 {4,4, 3, 5} 32 I 60 30 I 4. Concluding Remarks The graphs uniform with respect to M(v) are vertex-regular of special types. Some of them are vertex-symmetric1> but the others are not. A graph can be uniquely determined for a given sequence M(v), if one exists, except for M(v) = {4, 4, 4, 3}. Two non-isomorphic graphs can be obtained for M(v) ={4, 4, 4, 3}. It is interesting that one of them is vertex-symmetric, but the other is not. Besides regular polyhedrons, there are only four plane graphs, which are uniform with respect to W(e). They are edge-regular,0 and are uniquely determined for W(e). Uniform Plane Graphs 503 W(e) was used to partition the set of edges in the algorithm to test the iso-morphism or automorphism of planar graphs.2i M(v) can also be used for the test. The above result on M(v), however, shows that only M(v) is not enough for parti-tioning the vertices to test isomorphism, since the vertices in the non-isomorphic graphs of Fig. 9 have the same M(v) and can not be partitioned by use of M(v) only. Acknowledgements The authors are thankful for the encouragement and useful discussions given by Prof. Akira Kishima, Dept. of Electrical Engineering, Kyoto University. This work was supported in part by the Grant in Aid for Scientific Research of the Ministry of Education, Science and Culture of Japan under Grant: Cooperative Research (A) 135017 (1976). References 1) F. Harary, Graph Theory, Addison-Wesley, Mass., (1969). 2) J. Hopcroft and R. Tarjan, 'Isomorphism of Planar Graphs,' in Complexity of Computer Com. putations, Edit. R. Miller and J. Thatcher, Plenum Press, N. Y., (1972). 

Uniform Plane Graphs

https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/281051/1/mfeku_39_4_495.pdf

Uniform Plane Graphs

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository