Recognition of essentially disconnected benzenoids

Vol.8 No.4 ACTA MATHEMATICAE APPLICATAE SINICA Oct., 1992

S t u d y Bul le t in

RECOGNITION OF ESSENTIALLY

DISCONNECTED BENZENOIDS"

CHS RoNos, (S

( ColteCe o! Fi, lm,,~e =,ul E,:o,~,-,~8, F,u~&ou U.i,,ereitU, Fu.,bou SSO00~, Chi, m)

S.J . CYVlN and B .N. CYVIN

(The U,,~,,er~v of ~n~i~ Noway)

1. I n t r o d u c t i o n a n d D e f i n i t i o n s

A benzenoid, or a benzenoid system, is a connected planar graph whose every interior face is a regular hexagon. A peak (resp. valley) of a benzenoid is a vertex which lies above (reap. below) all its first neighbors. A Kekul4an benzenoid is a benzenoid with at least one perfect matching. An essentially disconnected benzenoid is a Kekul4an benzenoid which has some fixed bonds[I]. Essentially disconnected benzenoids have proved to be very useful in certain enumeration techniques for perfect matching[ 2[. Hence the problem of recognizing essentially disconnected benzenoids has attracted many researchers and several scientific papers have appeared{ 3-9].

In the search for criteria of essentially disconnected benzenoicls, the concept of horizontal g-cut plays an important role. A horizontal g-cut is either a horizontal cut or a horizontal proper g-cut; both of them are defined as follows[6]:

Def in i t i on 1[ 5] . A collection of edges C is called a horizontal cut of a benzenoid H if all the edges of C are intersected by a straight llne segment PIP2 which satisfies:

(1) P1P2 is horizontal and orthogonai to one of the three edge directions of H; (2) Every point of P1P2 is either an interior or a boundary point of some hexagon of

H; (3) Each of/ '1 and P2 is the center of a vertical edge of H; (4) The graph obtained from H by deleting all the edges of C has exactly two compo-

nents, called the upper bank and the lower bank, respectively (see Fig. la). De f in i t i on 2[ s]. A collection of edges C is called a horizontal proper g-cut of a

benzenoid H if all the edges of C are intersected by a broken line segment 1='1t>2t>3 which satisfies:

(i) P1P2 is horizontal and orthogonal to one of the three edge directions of H;

*Received January 9, 1990. Revised March 18, 1991.

378 ACTA MATHEMATICAE APPLICATAE SINICA Vol.8

(2) Each of P1 and P3 is the center of an edge lying on the contour of H, and P2 is the center of a hexagon of H;

(3) Every point of P1P2P3 is either an interior or a boundary point of some hexagon of H;

(4) The angle P1P2P3 is ~-/3; (5) The graph obtained from H by deleting all the edges of C has exactly two compo-

nents, called the upper and the lower banks, respectively (see Fig. lb).

Fig. la

upper bank

lower bank

.lower bank

Fig. lb

upper bank

The authors of [5] developed a method called the g-cut method to determine whether or not a given benzenoid is essentially disconnected. The method is based on the following theorem.

T h e o r e m 3[s]. Let H be a Kekul~an benzenoid. Then H is essentially disconnected if and only if there is at least one horizontal g-cut (for some position) such that s = t holds, where s is the number of edges intersected by the horizontal line segment PIP2, t is the difference between the number of peaks and the number of valleys lying in the upper bank of the horizontal g-cut.

The main purpose of this paper is to give a new criterion of essentially disconnected benzenoids which is better than that one described in Theorem 3, and to develop a rapid way to recognize essentially disconnected bensenoids based on an algorithm which enables us to determine the correspondence between the set of peaks and the set of valleys [l°].

2. A N e w C r i t e r i o n o f E s s e n t i a l l y D i s c o n n e c t e d B e n z e n o i d s

It is clear that a horizontal g-cut separates an essentially disconnected benzenoid into two parts: the upper bank and the lower bank. These two banks may not be complete benzenoids, namely, they may have some vertices of degree one {1].

D e f i n i t i o n 4. A standard horizontal g-cut is a horizontal g-cut with one of the corresponding banks being a complete benzenoid (see Fig. 1).

T h e o r e m 5. Let H be a Kekul~a~n benzenoid. Then H is essentially disconnected if and only if H has at least one standard horizontal g-cut for some position such that s -- t holds, where s and t have the same meaning as in Theorem 3.

Proo£ The sufficiency is obvious. In the following we prove the necessity. Suppose that H is an essentially disconnected benzenoid. Let C be a horizontal g-cut of H satisfying s -- t for some position. If C is standard, there is nothing to prove. Now assume that C is non-standard. Furthermore, we may assume that C is a horizontal proper g-cut (The case where C is a horizontal cut can be dealt with in a similar way). Then the lower bank (~orresponding to C is not a complete benzenoid. This implies that after deleting the edges intersected by line segment P1P2 or P2Ps, say P1P2, some edges which do not belong to any

No.4 ESSENTIALLY DISCONNECTED BENZENOIDS 379

hexagon of the lower bank will appear (see Fig. 2). According to the number of such edges, we distinguish three cases as follows:

Case 1. There is only one edge ex not belonging to any hexagon of the lower bank of C (see Fig. 2a).

Case 2. There are more tha~a one such edges (see Figs. 2b, 2c, 2d).

Subcase 2.1. One of these edges (say el) has an end vertex being of degree one (see Fig. 2b).

Subcase 2.2. None of these edges has an end vertex being of degree one (see Figs. 2c, 2d).

Case 3. All the edges below line segment t>1t>2 do not belong to any hexagon of the lower bank (see Fig. 2e).

For each case we shall find another horizontal g-cut C * such that both of its two banks have perfect matching (see Fig. 2). Note that for a horizontal g-cut the condition ~both of its two banks have perfect .matching ~ ~ equivalent to the condition ~it satisfies s -- t" (cf. the proof of Theorem 6 in {5]). Therefore, C' is another horizontal g-cut satisfying s' = t ~. As long as C ~ is not standard, another, horizontal g-cut will be found by arguments similar to the above. It is clear that at each stage the newly found horizontal g-cut is different from all those that have been found before. Since H is finite, it cannot have infinite number of horizontal g-cuts satisfying s -- t. This implies that we will eventually find a standard horizontal g-cut. The proof is complel ed.

Fig. 2a Fig. 2b

Fig. ~ el is single


Fig. 2d el is double

Fig. 2e

Since only a very small part of the horizontal g-cuts is standard, the above theorem is much better than Theorem 3 in the sense that it only needs to check standard horizontal g-cuts.

3. A N e w M e t h o d to Recognize Essent ia l ly D i s c o n n e c t e d B e n z e n o i d s

Recall that a peak-to-valley path is a monotonous path joining a peak with a valley. A perfect peak-to-valley path system is a set of mutually disjoint peak-to-valley paths containing all peaks and valleys of a benzenoid H[ 11]. For each Kekuldan benzenoid drawn in a fixed position there is a one-one correspondence between the set of perfect matching and the set of perfect peak-to-valley path systems[Ill.

It is not difficult to see that a perfect peak-to-valley path system determines a correspondence between all peaks and all valleys. Furthermore,. this correspondence does not depend on the choice of a perfect peak-to-valley path system[l°l. Thus we may assume that peak pi corresponds to valley vi under this correspondence. Let [piv~] denote the intersection graph induced by the vertices belonging simultaneously to the wetting region of p~ and the catchment region of vi (cf. [12]).

T h e o r e m 6. Let H be a Kekuldan benzenoid. Then H is essentially disconnected if and only if there are some edges that do not belong to any intersection graph [p~v~] for some position.

Proof. Necessity. Suppose that H is essentially disconnected. Then H has some fixed bonds. Let e be any

No.4 ESSENTIALLY DISCONNECTED BENZENOIDS 381

one of them. If e is a fixed single bond, then we draw hr in the plane so that e is nonvertical. If e is a fixed double bond, we d r a w / f in the plane so that e is vertical. Bear in mind that in any perfect peak-to-valley path system a non-vertical edge is a double bond, whereas a vertical edge is a single bond. Therefore, e does not belong to any perfect peak-to-valley path system for some position. Consequently, e does not belong to any intersection graph [pivl] for that position.

Sufficiency. Suppose that/I has some edges that do not belong to any intersection graph [piv~] for

some position. Let e be any one of them. Then e does not belong to any perfect peak- to-valley path system. Since there is a one-one correspondence between the set of perfect matching and the set of perfect peak-to-valley path systems as mentioned above, let K be any perfect matching of/I, and P be the corresponding perfect peak-to-valley path system. Bear in mind that any vertical edge that does not belong to P is a double bond of K, whereas any non-vertical edge that does not belong to P is a single bond of K. Therefore, if e is vertical, e is a double bond of K. By the arbitrariness of the perfect matching K, e is double in any perfect matching of H, which implies that e is a fixed double bond of h r. Similarly, e is a fixed single bond if e is non-vertical. Consequently, hr is essentially disconnected. The proof is thus completed.

By virtue of the above theorem we develop a new rapid way to recognize essentially disconnected benzenoids, called the P-V method: Draw a given benzenoid in a fixed position such that one of the three edge directions is vertical; set up the correspondence between the peaks and the valleys (cf. [10]); figure out the intersection graphs [p~] . For example, the benzenoid drawn in Fig. 3 is essentially disconnected since there axe some edges not belonging to any intersection graph [piv~ 1 for i = 1, 2, 3, 4.

PI

~4 ~2

Fig. 3

For an arbitrary benzenoid hr, in order to decide whether or not /~ is essentially disconnected, there are two steps to take. Step one: Determine whether or n o t / f is Kekul6an. Step two: I f / t is Kekul6an, decide whether or not / - / is essentially disconnected. Hence in the case when it is not clear whether or not a given benzenoid is Kekuldan, the g-cut method can not be used directly. Otherwise, a false conclusion may result. A glance at the benzenoid depicted in Fig. 4 win indicate why this is so. One can easily find a standard cut for the benzenoid in Fig. 4, but it is not essentially disconnected since it is non-Kekul6an (cf. Theorem 6 in [13]). The P-V method has the advantage that we need not to know


whether or not a given benzenoid is Kekul~an in advance. Therefore, the P-V method is much more rapid than the g-cut method in the sense that we are able to combine the two steps of determining whether or not a given benzenoid is essentially disconnected into one procedure.

Fig. 4

A c k n o w l e d g e m e n t : The au thors would like to t h a n k the referee for t he va luab le commen t s .

R e f e r e n c e s

[I] S.J. Cyvin, B.N. Cyvin and I. Gutman, Number of Kekul~ Structures of Five-Tier Strips, Z. Natur-

forsch, 40a (1985), 1253. [2] S.J. Cyvin and I. Gutman, Kekul~ Structures in Benzenoid Hydrocarbons, Lecture Notes in Chemistry

Vo|.46, Sprir~er, Berlin Heidelberg, New York, 1988. [3] S.J. Cyvin and I. Gutman, Topological Properties of Bermenoid Hydrocarbons, Prat XLIV, Obvious

and Concealed Non-Kekul~an B~n~noids, J. I~fol. Struct., 150 (1987), 157-169. [4] Zhang Fu Ji and Chen Rong Si, When Each H a x ~ n of a Hexagonal system Covers It, Discrete Applied

Mathematics, 30 (1991), 63-75. [5] Chen Rong Si, S.J. Cyvin and B.N. Cyvin, Recognition of Essentially Disconnected Benzenoids,/~J'ah:h,

:IS (1990), 71. [6] Li Xue Liang and Zhang Fu Ji, A Fast Algorithm to Determine Fixed bonds in Hexagonal Systems,

Match, ~t$ (1990), 151. [7] Zhang Fu Ji and Li Xue Liang, Hexagonal System without Fixed Double Bonds, Match, 25 (1990),

251. [8] He Wen Chen and He Wen Jie, Some Topoiog/cal Properties of Normal Benzenoids and Coronoids,

Match, 2s (1990), 22s.

[9] S.J. Cyvin and I. Gutman, Topological Properties of Bensenoid Systems, XXXL~, the Number of Kekul~ Structures of Bensenoid Hydrocarbons Containing a Chain of Hexagons, J. Seth. Chem. Soc.,

80, 443. [10] Guo Xiao Feng and Zhang Fu Ji, Recognition Kekul~an Benzenoid Systems by C-P-V Path Elimination,

J. l~.ath. Chem., S (1990), lS7. [II] H. Sachs, Perfect Matchinp in He0cagonal Systems, Combinatorica, 4:1 (1984), 89-99. [12] I. Gutman and S.J. Cyvin, A New Method for the Enumeration of Kekul~ Structures, Chem. Phys.

Letters, 136 (1987), 137. [13] Zhang Fu Ji, Guo Xiao Feng and Chen Rong Si, The Existence of Kekul~ Structures in a Benzenoid

System, in: Advances in the Theory of Benzenoid Hydrocarbons, Topics in Current Chemistry, 153. ed. L Gutman and S.J. Cyvin (Springer-Verlag, Berlin-Heidelberg, 1990), 181-193.

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Recognition of essentially disconnected benzenoids