Proc. Nati Acad. Sci. USA Vol. 78, No. 3, pp. 1329-1332, March 1981 Mathematics

Enumeration of structure-sensitive graphical subsets: Calculations (graph theory/combinatorics) R. E. MERRIFIELD AND H. E. SIMMONS Central Research and Development Department, E. I. du Pont de Nemours and Company, Experimental Station, Wilmington, Delaware 19898 Contributed by H. E. Simmons, October 6, 1980

ABSTRACT Numerical calculations are presented, for all Table 2. Qualitative properties of subset counts connected graphs on six and fewer vertices, of the -lumbers of hidepndent sets, connected sets;point and line covers, externally Quantity Branching Cyclization stable sets, kernels, and irredundant sets. With the exception of v, xIncreases Decreases the number of kernels, the numbers of such sets are al highly p Increases Increases structure-sensitive quantities. E Variable Increases K Insensitive Insensitive The necessary mathematical machinery has recently been de- Variable Variable veloped (1) for enumeration of the following types of special cr Decreases Increases subsets of the vertices or edges of a general graph: (i) indepen- p Increases Increases dent sets, (ii) connected sets, (iii) point and line covers, (iv;' ex- X Decreases Increases ternally stable sets, (v) kernels, and (vi) irredundant sets. In e Increases Increases order to explore the sensitivity of these quantities to graphical K Insensitive Insensitive structure we have employed this machinery to calculate some Decreases Increases explicit subset counts. (Since no more efficient algorithm is known for irredundant sets, they have been enumerated by the brute-force method of testing every subset for irredundance.) 1. e and E are always odd. The 10 quantities considered in ref. 1 have been calculated 2. p(G) E(G). for all connected graphs of up to six points. The results are given 3. ar(G) + &(G) constant for graphs with the same lVi. in Table 1 (begins on following page). Inspection of these results We conclude that these numbers are useful characterizations shows that, with the exception of kernels, these quantities are of graphical structure which should find application (for in- quite sensitive to various aspects of graphical structure. One stance) in the analysis of molecular structure. Indeed, earlier way of summarizing this structure sensitivity for graphs on a work (2, 3) has shown that the numbers of independent vertex given number n of points is to regard the path Pn as the ref- and edge sets are sensitive to chemically significant details of erence structure with the other graphs being derived from it molecular structure and that even bond-strength patterns by repeated application of the operations of branching and cy- within a molecule are well correlated with the numbers of in- clization. The response of these quantities to these operations dependent sets intersecting the different bonds. We are cur- is summarized in Table 2. rently exploring the application of the other quantities consid- A few additional empirical observations, for which no theo- ered here to molecular structure analysis. retical reason is evident, can be made on the basis of the num- bers given in Table 1. 1. Merrifield, R. E. & Simmons, H. E. (1981) Proc. NatL Acad. Sci. USA 78, 692-695. The publication costs of this article were defrayed in part by page charge 2. Merrifield, R. E. & Simmons, H. E. (1980) Theor. Chem. Acta 55, payment. This article must therefore be hereby marked "advertise- 55-75. ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. 3. Hosoya, H. (1971) BulL Chem. Soc. Jpn 44, 2332-2339.

1329 Downloaded by guest on October 1, 2021 1330 Mathematics: Merrifield and Simmons Proc. Nad Acad. Sci. USA 78 (1981) Table 1. Counts of graphical subsets*

anX p E K £ a X E K L cr,% p E K l a X E K L

1 0 2 2 1 1 2 1 1 0 1 0 1 31 6 32 31 5 6 26 969 .768 973 15 76

2 3 4 3 2 3 .2 2 1 1 1 2 32 21 22 31 5 26 13 16 5 17 4 15 2 .-.. 3 5 7 5 2 5 3 4 1 3 2 3 33 KJ1 23 25 29 4 25 11 19 3 21 5 12 4 / 4 8 7 3 4 4 8 4 7 3 4 34 ( 26 31 27 3 27 9 25 2 23 4 9 5u 8 11 9 3 9 5 7 2 5 2 5 35 22 26 27 5 27 12 20 4 17 3 13

6 2 9 12 9 2 9 4 8 1 7 3 4 36 XV 24 29 25 3 25 10 23 2 25 5 10

7 7 13 11 3 7 6 15 5 13 3 6 37 33 38 33 2 33 6 32 1 31 5 6 8L1 7 14 11 2 11 7 14 7 11 2 11 38 18 32 39 5 24 18 32 18 39 5 24 6 15 13 3 6 8 30 16 27 3 12 39 K 19 34 35 5 26 16 38 13 43 6 21 '10 X 5 16 15 4 5 10 61 41 57 3 22 40 @N 19 31 35 4 30 17 37 15 37 4 28 11cO 13 16 17 4 15 8 11 3 9 3 9 41 {j 18 26 39 6 20 16 32 14 41 6 19 12 17 21 17 2 17 5 16 1 15 4 5 42 20 34 31 4 26 15 44 10 45 5 19

13 14 18 15 3 15 7 13 2 11 3 7 43 22 37 33 3 26 13 47 8 49 6 17 14. 12 20 17 3 13 9 27 6 25 4 10 44 47 19 28 33 5 22 15 40 11 41 5 18 15 12 22 19 3 16 10 25 8 23 4 14 45 20 32 33 5 21 14 44 10 41 4 16

16$ 11 19 21 5 12 10 23 9 21 4 12 46 25 39 35 3 25 10 57 5 55 5 10

17 13 22 19 3 13 8 29 5 27 4 8 47 F 21 33 33 4 .25 14 43 9 49 7 18

18° 11 22. 21 5 16 11 22 11 21 5 16 48 % 20 30 27 4 27 14 48 8 45 4 17 19 10 23 25 5 13 12 51 24 49 5 16 49 4}19 .30 35 5 20 14 39 9 49 7 16

20w. 11 24 21 3 11 11 57 17 55 5 15 50 C 22 33 29 3 23 12 51 6 53 6 13 21 i 10 25 23 4 14 13 50 26 49 6 20 51 17 41 41 4 26 22 81 43 87 5 42

22. 11 27 23 2 17 13 52 25 51 6 24 52 K 16 38 43 6 21 21 76 42 91 7 29

23 t 10 23 21 4 12 12 53 20 51 5 18 53 17 43 41 5 28 21 82 41 93 7 37

24 9 27 25 4 10 15 110 59 109 7 25 54 17 34 39 5 18 19 85 33 85 5 31

25 10 28 25 3 10 14 116 52 115 7 25 55 16 32 41 6 19 20 77 36 85 6 33

26 9 25 23 4 9 14 115 45 113 6 26 56 4gr 17 34 39 5 18 18 88 29 97 7 23 27 9 28 25 3 15 16 107 65 107 8 31 57 18 36 33 4 21 17 100 22 101 6 23

28 8 29 27 4 8 18 231 140 231 9 37 58 7J17 38 39 5 21 19 86 31 99 8 26 29 8 30 27 3 12 19 226 154 227 :10 41 59K 17 38 43 5 21 20 81 39 89 6 34 30 7 31 29 4 7 22 473 341 475 12 54 60 20 34 31 4 26 15 44 10 45 5 19 Downloaded by guest on October 1, 2021 Mathematics: Merrifield and Simmons Proc. NatL Acad. Sci. USA 78 (1981) 1331

Table 1. (Cont.)

oX p 6 K t 0' p X E h l aTI,,x P 6 K* l cr p E K l 61 17 37 37 4 28 20 87 33 91 E 35 91 .6 42 39 44 18 2!2 203 68 207 7 39

62 20 43 39 3 26 17 97 26 107 c 28 92 .8 53 49 33 18 22 462 160 483 13 48 I 14 63 21 41 37 3 21 14 111 17 113 I9 18 93 .4 48 49 ;5 16 28 409 214 431 9 57 8 64 19 40 41 5i19 16 100 25 97 l5 20 94 5 47 43 .3 23 26 439 161 451 10 52

65 18 40 37 4 23 18 94 28 99 7 25 95 L5 46 43 44 15 224 448 145 465 11 43 11 66 17 34 35 5i21 18 92 25 97 78 26 96 L5 50 47 44 16 26 430 191 449 10 49 1. 67 20 37 33 3321 15 107 18 113 8 19 97 L3 43 53 7 13 228 386 221 417 9 64 68 15 31 49 8316 20 64 41 97 9 24 98 %yS 1.3 52 51 6 24 .32 381 263 431 10 68 69 18 38 35 4420 16 99 20 109 8 22 99 13 50 49 6 20 E31 388 245 423 9 68 1 70 16 42 47 516 22 193 84 201 7 38 100 13 49 51 6 19 L30 386 241 435 11 54

71 17 44 41 44 19 20 214 61 219 8 30 101 14 52 49 4 22 %31 398 242 429 9 73 wy I 72 19 45 41 3 19 18 225 53 235 110 29 102 13 47 49 6 16 '29 393 223 431 10 57

73 17 42 39 44 17 18 223 45 233 9 30 103 13 43 45 6 15 28 402 181 421 9 65

74 K 14 43 49 7 18 25 168 102 207 ]10 35 104 14 50 47 4 20 29 408 220 437 10 60

45 5 25 26 180 103 201 8 52 105 14 46 4 19 27 423 170 447 11 54 75 15 47 l 43 76 14 44 47 7 21 26 170 106 195 8 46 106 13 47 49 6 17 30 387 231 411 8 72

77 16 44 41 4 22 22 200 70 217 10 37 107 15 56 51 2 24 34 398 265 421 6 94 My 78 16 48 45 4 22 24 190 92 209 9 44 108 14 48 45 4 17 27 420 198 451 11 42

79 16 43 39 3 26' 23 201 72 209 8 43 109 15i53 49 3 23 28 417 215 451 12 62 1 4 80 16 42 39 4 18 21 206 64 219 9 31 110 14 50 47 4 20 30 405 224 425 8 70 4 81 15 44 45 5 20 25 181 97 195 7 49 111 14 44 41 4 16 26 433 156 449 10 50

82 15 42 43 5 21 24 185 81 205 9 43 112 13 52 49 4 14 32 892 463 929 12 69

83 14 39 49 7 16 24 169 96 201 9 40 113 12 54 53 6 17 36 844 562 893 10 91

84 19 52 47 2 27 21 206 79 227 12 47 114.1 13 54 51 4 17 34 874 509 913 11 83 I? 85 15 44 45 5 19 24 184 93 207 9135 115 12 51 53 6 13 34 851 516 899 11 83

86 14 35 45 7 15 24 170 86 177 6i54 116 141354 51 4 14 30 913 440 947 13 68

87 15 40 41 5 18 23 191 75 203 8 41 1177 133 48 45 4 13 30 917 357 945 13 66

88 16 46 43 4 20 23 194 86 205 8 41 1183 2 55 53 5 20 37 837 580 901 11 91

89 16 38 35 4 17 20 216 50 221 8332 119 2 53 51 5 18 36 846 538 891 10 96 90 IV%%L/ 16 49 45 3 24 27 185 104 197 6 61 120 / 12 47 45 5 15 32 886 382 917 12 79 Downloaded by guest on October 1, 2021 1332 Mathematics: Merrifield and. Simmons Proc. Nad Acad. -Sci.' USA 78'(1981)

Table 1. (Cont.) OYx p E K & a* p X 8 K l 0,X p E K l O p X 8 K &

121 g> 12 53 51 5 18 35 852 530 911 12 78 133 4 11 58 55 4 20 43 1,780 1,289 1,877 13 119

122 (N> 12 51 49 5 16 34 865 488 909 11 80 134 w 11 56 53 4 19 42 1,795 1,191 1,865 12 131 .123 * 13 57 53 3 22 38 849 581 895 9 110 135 10 59 57 5 11 48 3,740 2,698 3,851 14 156 124 %4r 12 54 55 5 20 36 835 576 913 13 78 136 11 60 57 4 11 46 3,799 2,585 3,891 15 151 125 %N 13 55 51 3 21 35 867 527 917 12 90 137 l 10 57 55 5 10 46 3,779 2,472 3,873 15 147 126 % 12 56 53 4 12 38 1,855 1,088 1,913 14 101 138 10 60 57 4 16 50 3,715 2,804 3,835 13 171 127 w 11 55 57 6 11 40 1,796 1,201 1,873 13 114 139 4 10 61 57 3 22 51 3,700 2,902 3,847 14 165 128 - 11 57 55 5 15 42 1,789 1,247 1,867 12 121 140 4 9 61 59 5 9 56 7,688 5,893 7,845 15 215

129 4 , 11 49 47 5 11 36 1,878 809 1,921 15 98 141 % 9 62 59 4 13 58 7,649 6,119 7,823 14 228 130 127 11 55 53 5 13 40 1,814 1,141 1,883 13 109 142 K6-e 8 63 61 5 8 66 15,645 12,936 15,877 15 300 131 J 12 59 55 3 18 42 1,812 1,240 1,881 12 129 143 K 7 64 63 6 7 76 31,738 27,449 32,057 15 406

132 * 11 58 55 4 20 44 1,774 1,297 1,857 11 136

* oa(&), number of independent vertex (edge) sets; p(p-), number of connected vertex (edge) sets; x(x-, number of vertex (edge) covers; E(g), number of externally stable vertex (edge) sets; K(K), number of vertex (edge) kernels; &(&, number of irredundant vertex (edge) sets. Downloaded by guest on October 1, 2021