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J. Chem. If: Comput. Sci. 1995, 35, 759-760
A Remark on the Naming of Cata-Condensed Benzenoids with Base 5 Numbers’ Wolfgang R. Muller, Klaus Szymanski, and Jan V. h o p Computer Center, The Heinrich Heine University, D-40225 Dusseldorf, The Federal Republic of Germany Nenad TrinajstiC* The Rugjer BoSkoviC Institute, P.O.B.1016, HR-41001 Zagreb, The Republic of Croatia Received March 14. 1995@ The naming of cata-condensed benzenoids with base 5 numbers as proposed by Elk is compared to the DAST code for polyhexes. As a result a counterexample to Elk’s algorithm is found. Recently Elk’ proposed in this journal an algorithm using base 5 numbers to designate canonical names to catacondensed benzenoid hydrocarbons. Since we have been using base 8 numbers for naming planar polyhexes-namely the DAST code-for some time we compared the two methods for the case of cata-condensed benzenoids. As their main difference, we found that Elk’s naming is not unique. A polyhex is a graph-theoretic abstraction for the carbon skeleton of a hydrocarbon made up of regular hexagons6 and as such represents a finite edge-connected subset of a hexagonal tesselation of the plane. A polyhex is a benzenoid graph if it is a 1-factorable graph.’ The term 1-factorization signifies that the corresponding benzenoid hydrocarbon possesses KekulC structure(s).8 A benzenoid graph (benzenoid hydrocarbon) is named cata-conden~ed,~.’~ if all vertices (carbon atoms) lie on its boundary, Le., if there are no inner vertices (that is, vertices belonging to three hexagons). I On the outer boundary of any polyhex there are six vertices more of order 2 than of order 3, so there must exist boundary edges between vertices of order 2. Thus, it is convenient to define a starting polyhex (P,a,b) as a polyhex P together with an ordered pair (a,b) of vertices of order 2 connected by a boundary edge. This pair is named the entrance of the starting polyhex. Let (P,a,b) be a starting polyhex and the vertex sequence (a,b,c,d,ef,a) describe a path around the starting hexagon. According to the definition there are only neighbor hexagons, namely NI, N2, and N3 possible beyond those edges not incident to a or b. This gives rise to a decomposition of the polyhex into at most four disjoint areas: Po-the starting hexagon itself; PI-the connectivity component of NI (if present) after elimination of PO,N2, and N3; Pz-the connectivity component of N2 (if present) after elimination of PO,PI,and N3; and P3-the connectivity component of N3 (if present) after elimination of PO,P I , and Pz. This decomposition will obviously not be independent of the ordering of the neighbors, so there must be an arbitrary, but then fixed convention about this sequel. To code the A correction to “A Simplified Algorithm Using Base 5 Numbers to Assign Canonical Names to Cata-Benzenoid Polybenzenes” By Seymour Elk [J. Chem. In$ Comput. Sci. 34, 637-640 (1994)l may be found on page 786 of this issue. Abstract published in Advance ACS Abstracts, June 1, 1995. @
0095-233819511635-0759$09.00/0
FIRST (4) f
ENTRANCE
/
’
SECOND (1)
bvd “
THIRD (2)
Figure 1. Ordering of directions with respect to the entrance edge
and their weights (in brackets). presence or emptiness of any of the components in the three bits of an octal digit there must also be a convention about the mapping. We found it most useful to look first between e andf with weight 4, then between d and e with weight 1, and lastly between c and d with weight 2. This gives at most three smaller starting polyhexes (Plf,e), (Pz,e,d), and (P3,d,c). The above is depicted in Figure 1. Now we can define the “DAST (dualist angle-restricted spanning tree) tuple” of a starting polyhex (with respect to the above convention) recursively: Let (P,a,b) be a starting polyhex, and then the decomposition above gives rise to at most three smaller starting polyhexes. We then construct the DAST tuple of (P,a,b) by adding the weights of the present neighbors to get a digit from 0 to 7 and appending to it the DAST tuples of the components (if present) ordered as the convention dictates. Because each decomposition produces smaller polyhexes, the recursion must finally terminate with cases with no remaining neighbors, which by the rule get the DAST tuple value of 0. The DAST code of a polyhex P is then defined as the lexicographic minimum among the DAST tuples of certain selected (corner-) entered polyhexes (P,a,b)(those for which after a suitable rotation and reflection a is the most northern among the most western vertices of the polyhex and b lies exactly south of a). In Figure 2 we give the DAST code corresponding to tribenzo[a,c,h]anthracene. In the case of cata-condensed polyhexes the DAST tuples can contain only the following five digits: 0, 1, 2, 4, and 6 . Giving up some generality by remapping these digits to 0, 1, 2, 3 and 4 we obtaine a base 5 number naming scheme for cata-condensed polyhexes. Elk’s approach, on the other hand, looks between d and e with weight 1, between e and f with weight 2 and between 0 1995 American Chemical Society
760 J. Chem. In$ Comput. Sei., Vol. 35,No. 4, 1995
121600
Figure 2. The DAST code belonging to tribenzo[a,c,h]anthracene.
(a) Unfiliar connections
MULLERET
4
AL.
1142030 (1 1 1 6 2 0 4 0 )
1142030 (1 1 1 6 4 0 2 0 )
0
(b) Polyfiliar connection
n
Figure 3. Elk’s convention for the orientations of the neighboring hexagons in regard to the starting benzene ring in a cata-condensed benzenoid.
Figure 5. Two nonisomorphic cata-benzenoids with identical base 5 numbers. The DAST codes, which are different, are given in brackets.
construction of counterexamples, Le., nonisomorphic polyhexes with the same canonical name. Perhaps the simplest case is given in Figure 5 , where two polyhexes are obviously nonisomorphic but get the same base 5 number by Elk’s algorithm. The conesponding DAST codes as expected possess different values. ACKNOWLEDGMENT
02400 2400 24
04200 4200 42
Figure 4. Elk‘s tuples using the base 5 number system for two orientations of benzo[g]chrysene. The canonical name is one with the lowest base 5 number. The DAST code for benzo[g]chrysene is 12600. c and d with weight 3. He calls these three orientations unfiliar connections to the starting hexagon.I2 It replaces 2 3 by 4 for the leading digit in the case of branching (he calls this one a p o l y f h r connection to the starting hexagon), and in this case orders the two subtuples lexicographically. This is shown in Figure 3. Elk selects as the canonical name the lexicographic minimum of the tuples for all starting polyhexes. However, he inconsistently writes a 0 instead of a first digit 1 (the latter would fit much more systematically to the leading 4 in the example in Figures 4 and 5 of Elk’s article). In Figure 4 we give as an example Elk’s tuples using the base 5 number system for two orientations of benzo[g]chrysene. The replacement of a fixed convention for the branched case by a lexicographic ordering opens Elk’s method to the
+
N.T. was supported in part by Ministry of Science and Technology of the Republic of Croatia through Grant No. 1-07-159. We thank Dr. G. W. A. Milne, the Editor, and the referees for their constructive comments. REFERENCES AND NOTES (1j Elk, S. B. A Simplified Algorithm Using Base 5 Numbers To assign Canonical Names to Cata-Benzenoid Polvbenzenes. J . Chem. Inf. Comput. Sci. 1994, 34, 637-640. Muller, W. R.; Szymanski, K.; Knop, J. V.; NikoliC, S.; TrinajstiC, N. On Counting Polyhex Hydrocarbons. Croat. Chem. Acta 1989, 62, 481 -483. Muller, W. R.; Szymanski, K.; Knop, J. V.; Nikolid, S.; TrinajstiC, N. On the Enumeration and Generation of Polyhex Hydrocarbons. J . Comput. Chem. 1990, 11, 223-235. Knop, J. V.; Muller, W. R.; Szymanski, K.; TrinajstiC, N. Use of Small Computers for Large Computations: Enumeration of Polyhex Hydrocarbons. J . Chem. In$ Comput. Sci. 1990, 30, 159-160. TrinajstiC, N.; NikoliC, S.; Knop, J. V.; Muller, W. R.; Szymanski, K. Computational Chemical Graph Theory: Characterization, Enumeration and Generation of Chemical Structures by Computer Methods; Simon & Schuster/Horwood: Chichester, 1991. TrinajstiC, N. Chemical Graph Theory, 2nd revised ed.; CRC Press: Boca Raton, FL. 1992; p 30. Knop, J. V.; Muller, W. R.; Szymanski, K.; TrinajstiC, N. On the Enumeration of 2-Factors of Polyhexes. J. Comput. Chem. 1986, 7, 547-564. Cyvin, S. J.: Gutman, I. Kekult Structures in Benzenoid Hydrocarbons: Springer-Verlag: Berlin, 1988. Balaban, A. T. Chemical Graphs. V. Proposed nomenclature of Branched Cata-Condensed Benzenoid Hydrocarbons. Tetrahedron 1969, 25, 2949-2956. Trinajstid, N. On the Classification of Polyhex Hydrocarbons. J . Math. Chem. 1990, 5, 171-176. Dionova-Jerman-BlaiiE, B.; TrinajstiC, N. Computer-Aided Enumeration and Generation of the KekulC Structures in Conjugated Hydrocarbons. Comput. Chem. 1982, 6 , 121-132. Taylor, F. L. Enumerative Nomenclature for Organic Ring Systems. Ind. Eng. Chem. 1948, 40, 734-738.
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