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Counting Structures in Grid Graphs, Cylinders and Tori Using Transfer Matrices: Survey and New Results (Extended Abstract)
, 2005
"... There is a very large literature devoted to counting structures, e.g., spanning trees, Hamiltonian cycles, independent sets, acyclic orientations, in the n × m grid graph G(n, m). In particular the problem of counting the number of structures in fixed height graphs, i.e., fixing m and letting n grow ..."
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There is a very large literature devoted to counting structures, e.g., spanning trees, Hamiltonian cycles, independent sets, acyclic orientations, in the n × m grid graph G(n, m). In particular the problem of counting the number of structures in fixed height graphs, i.e., fixing m and letting n grow, has been, for different types of structures, attacked independently by many different authors, using a transfer matrix approach. This approach essentially permits showing that the number of structures in G(n, m) satisfies a fixeddegree constantcoefficient recurrence relation in n. In contrast there has been surprisingly little work done on counting structures in gridcylinders (where the left and right, or top and bottom, boundaries of the grid are wrapped around and connected to each other) or in gridtori (where the left edge of the grid is connected to the right and the top edge is connected to the bottom one). The goal of this paper is to demonstrate that, with some minor modifications, the transfer matrix technique can also be easily used to count structures in fixed height gridcylinders and tori.
Counting Selfavoiding Walks in Some Regular Graphs
"... A selfavoiding walk (SAW, for short) is a path on a graph G that does not visit any node more than once. The problem of counting the SAWs in a given “regular ” graph G, such as the one shown in Figure 1, plays a crucial role in modeling many important problems in different areas of science, such a ..."
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A selfavoiding walk (SAW, for short) is a path on a graph G that does not visit any node more than once. The problem of counting the SAWs in a given “regular ” graph G, such as the one shown in Figure 1, plays a crucial role in modeling many important problems in different areas of science, such as combinatorics, statistical physics, theoretical chemistry, and computer science. By “twodimensional grid ” we mean the twodimensional rectangular lattice Z 2 with origin (0, 0). One of the most prominent applications of counting SAWs is the modeling of spatial arrangement of linear polymer molecules in a solution. Here a SAW represents a molecule composed of monomers linked together in a chain by chemical bonds. Other application areas include the percolation model, the Ising model, and the network reliability model. Valiant [Val79b] is the first to find connections between the problem of counting SAWs and computational complexity theory. He showed that the problem of counting SAWs between two given points, the problem of counting Hamiltonian cycles, and the probFigure 1: A length18 in the complete 2D grid. lem of counting Hamiltonian paths between two given points are all #Pcomplete under polynomial parsimonious reductions (that is, polynomialtime reductions of functions not requiring postcomputation) both for directed graphs and for undirected graphs. One might ask for which types of graphs these counting problems remain #Pcomplete. The goal of this paper is to present some natural regular graphs for which the corresponding counting problems are #Pcomplete and discuss some open issues. 2