Polyomino

There is a literature (see, for example, [GrSh87a, Section 9.4], [BGOR99], [BCH02], and [BCH03]) about proper parabolic polycycles (polyhexes, polyamends, polyominoes for 6,3, 3,6), 4,4, respectively) the terms come from familiar terms hexagon, diamond, domino, where the last two correspond to the case p, ... 

Polyominoes were considered first by Conway, Penrose, Golomb as tiles (of R2 etc. see, for example, [CoLa90]) and in Game Theory later, they woe used for enumeration in physics and statistical mechanics. 

BGOR99] M. Bousquet-Mdlou, A J. Guttman, W.P. Qrrick, and A. Rechnitzer, Inversion Relations, Reciprocity and Polyominoes, Armais of Combinatorics 3 (1999) 223-249. 

CoLa90] J.H. Conway and J.C. Lagarias, Tilings with polyominoes and combinatorial group theory, Journal of Combinatorial Theory, Series A 53 (1990) 183-208. 

Har90] H. Harborth, Some mosaic polyominoes, Ars Combinatoria 29A (1990) 5-12. 

Fig. 4. All Clar colorings of T3(2, 2,2) and the corresponding distributions into important patterns in mathematical chemistry. K is the king pattern, M a dimer pattern, D a domino distribution, while P is a set of a special polyomino pattern. The set L is the set of matchings of L4. Any of the above distributions generate F4...

Chart. 1. Nonadjacency relations. 0(G, k) is a general relation which counts the number of k nonadjacent objects in G. p(T, k) = the number of k-matchings in T, o(A, k) is the number of selections of k nonadjacent vertices in A, r(B, k) is the fcth resonant number of B, [Pg.279]

Fig. 5. A Gutman tree, T, an unbranched benzenoid system, B, a Clar graph, A, a king polyomino-graphy, P, and a rook board, Pr...

Fig. 6. All 3-matchings in T4(2,1,0,1) and the corresponding object-distribution in other equinumerical graphs, viz., an unbranched benzenoid hydrocarbon, a king polyomino and a rook board...

A plane figure with many sides, polyomino... 

In this review we will focus on polyhex graphs, caterpillar trees, Clar graphs and several related polyomino gra s [19] In addition sets of graphs obeying certain types of recursive relations, called "Fibonacci Graphs" will be discussed particularly from the point of view of their computational importance ... 

Number of ways of arranging k non-taking kings on a polyomino graph. 

First we will focus attention on selected topics relating to the equivalence between benzenoid hydrocarbons, and special types of graphs and other mathematical objects that we can associate with benzenoids- In particular we will explore relations involving caterpillar trees [3] associated with catacondensed benzenoids and their line graphs [17] called, as already mentioned, Clar graphs [4]. Also relations involving "boards" (known technically as polyominos) of special properties such as those associated with "king" and "rook" pieces of chess... 

King polyomino [19] Benzenoid system containing no more than 4 hexagons... 

Two cells on a square are said to be adjacent it they share at least one corner Observe that the above dehnition of a polyomino does not determine the Aane of the board, but just the adjacency Hence in our example (Fig 7), P can be pictorially represented by tmy of the following apes ... 

Let us consider more closely the set of objects of Fig 7 associated with the same adjacency matrix A We will refer to these as the set T, A, B, P standing, respectively, for a caterpillar tree, a Clar graph (As L(T)) a benzenoid graph and a king polyomino The grai invariants that we will consider in each case are as follows ... 

Fig 11 A set of twelve gtaphs comprising four types a caterpillar, a polyhex graph, a Clar graph and a polyomino graph-... 

The term "pseudo" king polyomino is used here to indicate that they ate not proper polyomino graphs [37] but contain factors of the variable x The same terminology might be extended to caterpillars. See, S- El-Basil, J- Math- Chem-, i, 161 (1987) D-H- Rouvray smd S- El-Basil, J. Molecular Structure (Theochem-), 165. 13 (1988). 

Klarner DA (1965) Some Results Concerning Polyominoes. Fibonacci Quarterly 3 9... 

Lunnon WF (1972) Counting Hexagonal and Triangular Polyominoes [in] Graph Theory and Computing (Read RC, Edit). Academic Press, New York 87... 

Reddmeier DH (1981) Counting Polyominoes - Yet Another Attack. Discrete Math 36 191... 

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