ars mathematica contemporanea
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 10 (2016) 281-290
Cube-contractions in 3-connected quadrangulations
Yusuke Suzuki *
Department of Mathematics, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-2181, Japan

Received 4 October 2013, accepted 10 January 2016, published online 30 January 2016
Abstract
A 3-connected quadrangulation of a closed surface is said to be K' -irreducible if no face- or cube-contraction preserves simplicity and 3-connectedness. In this paper, we prove that a K'-irreducible quadrangulation of a closed surface except the sphere and the projective plane is either (i) irreducible or (ii) obtained from an irreducible quadrangulation H by applying 4-cycle additions to F0 C F(H) where F(H) stands for the set of faces of H. We also determine K'-irreducible quadrangulations of the sphere and the projective plane. These results imply new generating theorems of 3-connected quadrangulations of closed surfaces.
Keywords: Quadrangulation, closed surface, generating theorem. Math. Subj. Class.: 05C10
1 Introduction
In this paper, we only consider simple graphs which have no loops and no multiple edges. We denote the vertex set and the edge set of a graph G by V(G) and E(G), respectively. We say that S c V(G) is a cut of G if G - S is disconnected. In particular, S is called a k-cut if S is a cut with |S| = k. A cycle C of G is said to be separating if V(C) is a cut. Similarly, a simple closed curve 7 on a closed surface F2 is said to be separating if F2 - 7 is disconnected.
A quadrangulation G of a closed surface F2 is a simple graph cellularily embedded on the surface so that each face is quadrilateral; thus, a 2-path on the sphere is not a quadrangulation. We denote the set of faces of G by F(G) throughout the paper. For quadrangulations we consider applying three reductions, called a face-contraction, a 4-cycle removal
*This work was supported by JSPS Grant-in-Aid for Young Scientists (B) No. 24740056. E-mail address: y-suzuki@math.sc.niigata-u.ac.jp (Yusuke Suzuki)
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
282
Ars Math. Contemp. 10 (2016) 255-268
[voVi]!/
Avi
face-contraction
4-cycle removal
"t<V3
[V0W2]I/
Avi
cube-contraction
Figure 1: Reductions for quadrangulations.
and a cube-contraction, as shown in Figure 1. (Precise definitions of these reductions will be given in the next section.) The corresponding inverse operations are called a vertex-splitting, a 4-cycle addition and a cube-splitting, respectively. In particular, the operations of a face-contraction and a 4-cycle removal were first introduced by Batagelj [1]
Irreducible quadrangulations, such that no face-contraction is applicable without making a loop or multiple edges, on a fixed closed surface with low genus were obtained in earlier papers. In [9], it was proven that a 4-cycle is the unique irreducible quadrangulation of the sphere, and that there exist precisely two irreducible quadrangulations of the projective plane shown in Figure 2, where Qp and Qp are the unique quadrangular embeddings of K4 and K3j4 on the projective plane, respectively. The irreducible quadrangulations of the torus and the Klein bottle have also been determined in [6, 5]. In [8], it was proven that for any closed surface F2 there exist only finitely many irreducible quadrangulations of F2, up to homeomorphism.
A 3-connected quadrangulation G of a closed surface F2 is said to be K3-irreducible if any of a face-contraction and a 4-cycle removal breaks simplicity or 3-connectedness of G. The following theorem is the starting point of the study of 3-connected quadrangulations. (The definitions of a pseudo double wheel, a Mobius wheel and a double cube are given in the next section.)
Theorem 1.1 (Brinkmann et al.[2]). Any K3-irreducible quadrangulation of the sphere is isomorphic to a pseudo double wheel.
Observe that a 3-connected quadrangulation of the sphere corresponds to a 4-regular 3-connected graph on the same surface by taking its dual. Broersma et al. [3] considered the same problem of the dual version with weaker conditions than those of Brinkmann. For the projective plane, Nakamoto proved the following.
Theorem 1.2 (Nakamoto[7]). Any K3-irreducible quadrangulation of the projective plane is isomorphic to either a Mobius wheel or Qp.
Furthermore, the results in [4] imply the following.
Y. Suzuki: Cube-contractions in 3-connected quadrangulations
283
Theorem 1.3 (Nagashima et al.[4]). Let G be a quadrangulation of a closed surface other than the sphere and the projective plane. Then G is K3-irreducible if and only if G is irreducible.
Figure 2: Irreducible quadrangulations on the projective plane.
In this paper, we determine other minimal subsets of 3-connected quadrangulations by replacing 4-cycle removals with cube-contractions. A 3-connected quadrangulation G is said to be K3-irreducible if any of a face-contraction and a cube-contraction breaks the simplicity or the 3-connectedness of G. The followings are our main results in the paper. In these statements, F(H) stands for the set of faces of a quadrangulation H.
Theorem 1.4. Let G be a K'3-irreducible quadrangulation of a closed surface F2 other than the sphere and the projective plane. Then, G is either (i) irreducible or (ii) obtained from an irreducible quadrangulation H by applying 4-cycle additions to F0 C F(H).
Theorem 1.5. Let G be a fc3-irreducible quadrangulation of the sphere. Then, G is either (i) a pseudo double wheel or (ii) a double cube.
Theorem 1.6. Let G be a fc3-irreducible quadrangulation of the projective plane. Then, G is (i) a Mobius wheel, (ii) Qp or (iii) obtained from Qp (resp. Qp) by applying 4-cycle additions to F0 C F(Qp) (resp. F0 C F(Qp)).
Corollary 1.7. For any closed surface F2, there exist only finitely many quadrangulations which are K'3-irreducible but are not K3-irreducible, up to homeomorphism.
This paper is organized as follows. In the next section, we define the reductions used in this paper and introduce typical 3-connected quadrangulations on the sphere and the projective plane called a pseudo double wheel and a Mobius wheel, respectively. In Section 3, we develop some theoretical tools and prove Theorem 1.4. The last section is devoted to prove the planar case and the projective-planar case individually, using some figures.
2 Reductions and typical quadrangulations
Let G be a quadrangulation of a closed surface F2 and let f be a face of G bounded by a cycle v0viv2v3. (We also use the notation like f = v0v1v2v3 in this paper.) The face-contraction of f at {v0, v2} in G consists of identification of v0 and v2, and replacement of the resulting multiple edges {v0vi, v2vi} and {v0v3, v2v3} with two single edges, respectively. In the resulting graph, let [v0v2] denote the vertex arisen by the identification of v0
284
Ars Math. Contemp. 10 (2016) 255-268
and v2 (see the left-hand side of Figure 1). Similarly, we define the face-contraction of f at jvi, v3}. The inverse operation of a face-contraction is called a vertex-splitting. We say that f is contractible at {v0, v2} in G, if the graph obtained from the face-contraction of f at {v0, v2} is simple. Assume in addition that G is 3-connected. A face f of G is said to be
3-contractible	at {v0, v2} if f is contractible at {v0, v2} and the graph obtained from the face-contraction is still 3-connected.
Let f = v0v1v2v3 be a face of a quadrangulation G of F2. A 4-cycle addition to f consists of inserting a 4-cycle C = m0m1m2m3 inside f in G and joining v and u for i = 0,1,2, 3. The inverse operation of a 4-cycle addition is called a 4-cycle removal (of C), as shown in the center of Figure 1. We call the subgraph Q isomorphic to a cube with eight vertices «j, v for i = 0,1,2, 3 an attached cube. For an attached cube Q, we call the above 4-cycle C an inner 4-cycle of Q. In addition, we denote dQ = v0v1v2v3. Let G be a 3-connected quadrangulation of a closed surface having an attached cube Q. We say that an inner 4-cycle C of Q (or easily an attached cube Q) is removable if the graph obtained from G by applying 4-cycle removal C preserves the 3-connectedness. (Observe that a 4-cycle removable never destroy simplicity of G.)
As mentioned in the introduction, there exist some results of 3-connected quadrangu-lations (or quadrangulations with minimum degree 3) on surfaces. In those results, the
4-cycle	removal is necessary by the following reason: Let G be the graph obtained from a 3-connected quadrangulation G of a closed surface by applying 4-cycle additions to all faces of G. Clearly G is 3-connected, but we cannot apply any face-contraction to G without creating a vertex of degree 2.
Our third reduction of quadrangulations of closed surfaces is defined as a sequence of the above two reductions. Assume that a quadrangulation G has an attached cube Q with an inner 4-cycle C and with dQ = v0v1 v2v3. A cube-contraction of Q at {v0,v2} in G consists of a 4-cycle removal of C followed by a face-contraction at {v0, v2} (see the right-hand side of Figure 1). The inverse operation of a cube-contraction is called a cube-splitting. We say that an attached cube Q is contractible if the graph obtained from G by applying a cube-contraction of Q preserves the simplicity and the 3-connectedness. One might suspect that if an attached cube Q is contractible then Q is removable (and the face that appeared by the removal is contractible). However, this is not true in general since a 4-cycle removal might break the 3-connectedness of the graph.
Figure 3: W8 and W5.
We need to describe two special types of embeddings. Firstly, embed a 2k-cycle C = v0M0v1u1... vk-1 wfc-1 (k > 3) into the sphere, put a vertex x on one side and a vertex y on
Y. Suzuki: Cube-contractions in 3-connected quadrangulations
285
the other side and add edges xv and yu for i = 0,..., k —1. The resulting quadrangulation of the sphere with 2k + 2 vertices is said to be apseudo double wheel and denoted by W2k (see the left-hand side of Figure 3). The smallest pseudo double wheel is W6, which is isomorphic to a cube, when the graphs are assumed to be 3-connected. The cycle C of length 2k is called the rim of W2k. We call a quadrangulation of the sphere obtained from W6 by a single 4-cycle addition a double cube, which is isomorphic to C4 x P2.
Secondly, embed a (2k -1)-cycle C = v0vi... v2k-2 (k > 2) into the projective plane so that the tubular neighborhood of C forms a Mobius band. Next, put a vertex x on the center of the unique face of the embedding and join x to v for all i so that the resulting graph is a quadrangulation. The resulting quadrangulation of the projective plane with 2k vertices is said to be a Mobius wheel and denoted by TW2fc-1 (see the right-hand side of Figure 3).
3 Lemmas to prove Theorem 1.4
The following lemma holds not only for quadrangulations but also for even embeddings of closed surfaces F2, that is, for graphs embedded on F2 with each face bounded by a cycle of even length. Taking a dual of an even embedding and using the odd point theorem, we can easily obtain this lemma.
Lemma 3.1. An even embedding of a closed surface has no separating closed walk of odd length.
Let G be a quadrangulation of a closed surface F2 and let f = v0v1 v2v3 be a face of G. Then a pair {vj, vi+2} is called a diagonal pair of f in G, where the subscripts are taken modulo 4. A closed curve 7 on F2 is said to be a diagonal k-curve for G if 7 passes only through distinct k faces f0,..., fk-1 and distinct k vertices x0,..., xk-1 of G such that for each i, f and fi+1 share xj, and that for each i, {xi-1, x4} forms a diagonal pair of f of G, where the subscripts are taken modulo k.
Lemma 3.2. Let G be a quadrangulation of a closed surface F2 with a 2-cut {x, y}. Then there exists a separating diagonal 2-curve for G only through x and y.
Proof. Observe that every quadrangulation of any closed surface F2 is 2-connected and admits no closed curve on F2 crossing G at most once. Thus there exists a surface separating simple closed curve 7 on F2 crossing only x and y, since {x, y} is a cut of G.
We shall show that 7 is a diagonal 2-curve. Suppose that 7 passes through two faces f1 and f2 meeting at two vertices x and y. If 7 is not a diagonal 2-curve, then x and y are adjacent on df1 or df2. Since G has no multiple edges between x and y, and since {x, y} is a 2-cut of G, we may suppose that x and y are adjacent in df1, but not in df2. Here we can take a separating 3-cycle of G along 7. This contradicts Lemma 3.1.	□
Lemma 3.3. Let G be a 3-connected quadrangulation of a closed surface F2, and let f = v0v1v2v3 be a face of G. If the face-contraction of f at {v0, v2} breaks 3-connectedness of the graph but preserves simplicity, then G has a separating diagonal 3-curve passing through v0,v2 and another vertex x G V(G) — {v0, v1, v2, v3}.
Proof. Let G' be the quadrangulation of F2 obtained from G by the face-contraction of f at {v0, v2}. Since G' has connectivity 2, G' has a 2-cut. By Lemma 3.2, G' has a separating diagonal 2-curve 7' passing through two vertices of the 2-cut. Clearly, one of the two
286
Ars Math. Contemp. 10 (2016) 255-268
vertices must be [v0v2] of G', which is the image of v0 and v2 by the face-contraction of f. (Otherwise, G would not be 3-connected, a contradiction.) Let x be a vertex of G' on Y' other than [v0v2]. Note that x is not a neighbor of [v0v2] in G'. Now apply the vertex-splitting of [v0v2] to G' to recover G. Then a diagonal 3-curve for G passing through only v0,v2 and x arises from y' for G'.	□
The next lemma plays an important role in a later argument.
Lemma 3.4. Let G be a 3-connected quadrangulation on a closed surface F2. If G has a separating 4-cycle C = x0x1x2x3 and a face f of G such that
(i)	one of the diagonal pairs of f is {x^ xi+2} for some i, and
(ii)	f has a separating diagonal 3-curve y intersecting C only at xi and xi+2 transversely,
then there exists a 3-contractible face in G.
Proof. Suppose that G has a separating 4-cycle C = x0xix2x3 and a face f bounded by axicx3. Since C is separating, G has two subgraphs GR and GL such that GR U GL = G and Gr n Gl = C. Suppose that f is contained in GR. Furthermore, we assume that GR contains as few vertices of G as possible.
Since C is separating, we have df = C. By (ii), f has a separating diagonal 3-curve Y through x1, x3 and some vertex x. Note that x G V (GL) - V (C) by the condition (ii) in the lemma. Now assume that f is not 3-contractible at {a, c}. Observe that y (or the 3-cut {x1, x, x3}) separates a from c. Further, G does not have both of edges ax and cx since df = C. Therefore, there is no path of G of length at most 2 joining a and c other than ax1c and ax3c. Moreover, if {a, c} n {x0, x2} = 0, then f has no separating diagonal 3-curve joining a and c. This contradicts our assumption by Lemma 3.3 and so we may suppose that a = x0 and c = x2, and f has a separating diagonal 3-curve, say y', through a (= x0 ) and c.
Since y' separates x1 and x3 and since x2 is a common neighbors of x1 and x3, y' must pass through x2, and hence we can find a face f' of GR one of whose diagonal pair is {c, x2}. Let C' be the 4-cycle x1x2x3c of G. Since deg(c) > 3, we have df' = C', and hence C' is a separating 4-cycle in GR such that C' = C. Moreover, y' and C' cross transversely at x2 and c. Therefore, C' and f' are a 4-cycle and a face which satisfy the assumption of the lemma, and moreover, C' can cut a strictly smaller graph than GR from G. Therefore, this contradicts the choice of C.	□
Lemma 3.5. Let G be a 3-connected quadrangulation of a closed surface F2. If G is K3-irreducible then G is K3-irreducible.
Proof. Let G be a 3-connected quadrangulation of a closed surface. Assume that G is not K3-irreducible. Then, G has either a 3-contractible face or a contractible cube. If G has a 3-contractible face, then G is not K3-irreducible. Therefore, we suppose that G has no 3-contractible face but has a contractible cube Q with an inner 4-cycle C in the following argument.
Now, we apply a 4-cycle removal of C to G and let G' be the resulting quadrangulation. Let f' = dQ be the new face of G' into which C was inserted. If G' is 3-connected, G is not K3-irreducible by the definition, and we are done. Therefore, we assume that G' is not 3-connected. By Lemma 3.2, there is a diagonal 2-curve y passing through f' and another
Y. Suzuki: Cube-contractions in 3-connected quadrangulations
287
face f''; otherwise, G would have a 2-cut, contrary to our assumption. Note that f'' is also a face in G. Now dQ and f" satisfy the conditions of Lemma 3.4, and hence there exists a 3-contractible face in G. However, this contradicts the above assumption. Thus, the lemma follows.	□
In the following argument, we denote the set of K3-irreducible (resp. K3-irreducible) quadrangulations of a closed surface F2 by K3I(F2) (resp. K31(F2)).
Lemma 3.6. Let G be a 3-connected quadrangulation of F2. If G G K3I(F2) \K3I(F2), then G has an attached cube Q such that the graph obtained from G by applying a 4-cycle removal of Q is in K3I(F2).
Proof. Let G be in K3I(F2) \ K3I(F2). By the definition, G has an attached cube Q with an inner 4-cycle C which is removable, but is not contractible. We apply a 4-cycle removal of C and let G- be the resulting quadrangulation. We denote the new face of G- by f-, where f- = dQ.
First, we confirm that G- is 3-connected. Otherwise, G- has a 2-cut and has a separating diagonal 2-curve 7 on F2 by Lemma 3.2. If 7 does not pass through f- then 7 would also be a diagonal 2-curve in G, a contradiction. Let f0 be the other face passed by 7. Here, fo and dQ in G satisfy the conditions in Lemma 3.4 and there exists a 3-contractible face, contrary to G being K3-irreducible.
By way of contradiction, assume that G is not in K3I(F2). That is, G has either (a) a 3-contractible face or (b) a contractible cube. First, we assume (a) and let f be a 3-contractible face in G-. If f- = f, the attached cube Q in G would be contractible, contrary to G being K3-irreducible. Thus, suppose f- = f. In this case, let G' be the resulting 3-connected quadrangulation after applying a face-contraction of f in G-. Since any 4-cycle addition doesn't break the 3-connectedness of a quadrangulation, the graph obtained from G' by a 4-cycle addition to f- is clearly 3-connected. This means that f is also 3-contractible in G, a contradiction.
Next, suppose (b) and let Q' be such a contractible cube with dQ' = v0viv2v3. If Q' does not contain f- as one of its five faces, Q' is also contractible in G and G would not be K3-irreducible by the similar argument as above. Thus, we assume that Q' contains f-. Let C = m0m1m2m3 denotes the inner 4-cycle of Q' where WjVj G E(Q') for i = 0,1,2,3. We consider the following two cases up to symmetry; (b-1) f- = C and (b-2) f- = v0u0u1v1. At first, suppose (b-1). Here, we apply a face-contraction of f1 = v0u0u1v1 at {u0, v1} to G. If the above face-contraction breaks the 3-connectedness of G, there exists a face f2 = v1xv3y in the outside of Q' by Lemma 3.3; note that it clearly preserves the simplicity of the graph since v1 = v3. Now, a separating diagonal 3-curve passing through {v1, u0, v3} satisfies the conditions of Lemma 3.4 and hence G is not K3-irreducible, contrary to our assumption. In fact, an analogous proof is valid for (b-2) if we try to apply a face contraction at {v1, u2} to G. Therefore the lemma follows.	□
Lemma 3.7. Let G be a 3-connected quadrangulation of a closed surface F2. If G G K3I (F2) \ K3I (F2), then G can be obtained from H G K3I (F2) by applying 4-cycle additions to F0 C F(H).
Proof. Assume that G G K3I(F2) \ K3I(F2). By the previous lemma, there exists a sequence of K3-irreducible quadrangulations G = G0, G1,..., Gk such that Gi+1 is obtained from Gj by a single 4-cycle removal of Cj, where Gk G K3I(F2). (Since the
288
Ars Math. Contemp. 10 (2016) 255-268
number of vertices of G is finite, Gk € K3I(F2).) Let Qi denote an attached cube in Gi with an inner 4-cycle Ci.
For a contradiction, we assume that there exists l € {0,..., k — 2} such that G; is obtained from G;+1 by a 4-cycle addition which is put on a face not of F(Gk); this l should be maximal. This implies that C; is put on a face of Q;+1 as one of its five faces. Then the same argument as the proof of Lemma 3.6 holds and hence G; would not be K'3-irreducible, contrary to our assumption. Thus for each i € {0,..., k — 1}, Gi is obtained from Gi+1 by a 4-cycle addition which is put on a face of F(Gk).	□
Proof of Theorem 1.4. By Lemma 3.5, we have K3I (F2) C K3I(F2). Furthermore, by Theorem 1.3 and Lemma 3.7, we obtain (i) and (ii) in the statement. Thus, we have got a conclusion.	□
4 Spherical and projective-planar cases
In this section, we discuss the spherical case and the projective-planar case.
Proof of Theorem 1.5. Let G be a K3-irreducible quadrangulation of the sphere. We have
K3I(S2) C K^I(S2) by Lemma 3.5, where S2 stands for the sphere.
If G is K3 -irreducible, then G is isomorphic to a pseudo double wheel by Theorem 1.1. If G is in K3I(S2) \ K31(S2), G can be obtained from a pseudo double wheel W2k (k > 3) by some 4-cycle additions to faces of W2k by Lemma 3.7. However if k > 4, G has a 3-contractible face (or a contractible cube), as shown in the first operation in Figure 4. (For example, the entire Figure 4 presents a sequence of a face-contraction and a cube-contraction which deforms W8 with an attached cube Q into W6, preserving the 3-connectedness.)
Figure 4: W8 with an attached cube Q deformed into W6.
Therefore, we only consider the case of k = 3 in the following argument. Assume that G is obtained from W6 by at least two 4-cycle additions to faces of W6. Similarly to the above argument, G would have a 3-contractible face (or a contractible cube) , as in Figure 5, contrary to G being K3-irreducible; note that it suffices to discuss these two cases, up to symmetry. Therefore, we conclude that G is obtained from W6 by exactly one 4-cycle addition. This is nothing but a double cube; observe that a double cube has no 3-contractible face and no contractible cube.	□
To conclude with, we prove the projective-planar case.
Y. Suzuki: Cube-contractions in 3-connected quadrangulations
289
Proof of Theorem 1.6. In this case, we use Mobius wheels Wk (k > 3) and Qp as base graphs by Theorem 1.2.
First we consider the former case. Similarly to the previous proof (and see Figure 6), we consider only a Mobius wheel W3 as a base to which we apply some 4-cycle additions. However, W3 (= Qp) is isomorphic to the complete graph with four vertices, and hence it is irreducible. This fact implies that every G obtained from W3 by applying at most three 4-cycle additions is K'3-irreducible since any face-contraction and any cube-contraction to G destroys the simplicity of the graph, or results in a vertex of degree 2. From this case, we obtain exactly three quadrangulations in K'3I(P2) \ K3I(P2), up to homeomorphism, where P2 stands for the projective plane.
Figure 6: W5 with an attached cube Q deformed into W3.
Similarly, as the latter case, we obtain the other ten quadrangulations in K31(P2) \ K3I(P2) from Qp; consider all the way to put attached cubes into faces of Qp, up to symmetry. As aresult, we have \K'3I(P2) \ K3I(P2)| = 13 in total.	□
290
Ars Math. Contemp. 10 (2016) 255-268
References
[1]	V. Batagelj, An inductive definition of the class of 3-connected quadrangulations of the plane, Discrete Math. 78 (1989), 45-53, doi:10.1016/0012-365X(89)90159-3.
[2]	G. Brinkmann, S. Greenberg, C. Greenhill, B. D. McKay, R. Thomas and P. Wollan, Generation of simple quadrangulations of the sphere, Discrete Math. 305 (2005), 33-54, doi:10.1016/j.disc. 2005.10.005.
[3]	H. J. Broersma, A. J. W. Duijvestijn and F. Gobel, Generating all 3-connected 4-regular planar graphs from the octahedron graph, J. Graph Theory 17 (1993), 613-620, doi:10.1002/jgt. 3190170508.
[4]	M. Nagashima, A. Nakamoto, S. Negami and Y. Suzuki, Generating 3-connected quadrangulations on surfaces, Ars Combin. 116 (2014), 371-384.
[5]	A. Nakamoto, Irreducible quadrangulations of the Klein bottle, Yokohama Math. J. 43 (1995), 125-139.
[6]	A. Nakamoto, Irreducible quadrangulations of the torus, J. Combin. Theory Ser. B 67 (1996), 183-201, doi:10.1006/jctb.1996.0040.
[7]	A. Nakamoto, Generating quadrangulations of surfaces with minimum degree at least 3, J. Graph Theory 30 (1999), 223-234, doi:10.1002/(SICI)1097-0118(199903)30:3(223::AID-JGT7)3.3. CO;2-D.
[8]	A. Nakamoto and K. Ota, Note on irreducible triangulations of surfaces, J. Graph Theory 20 (1995), 227-233, doi:10.1002/jgt.3190200211.
[9]	S. Negami and A. Nakamoto, Diagonal transformations of graphs on closed surfaces, Sci. Rep. Yokohama Nat. Univ. Sect. I Math. Phys. Chem. (1993), 71-97.