The number of realizations of all Laman graphs with at most 12 vertices
Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes, Josef Schicho
10.5281/zenodo.1245517


This data set consists of files for Laman graphs (minimally rigid graphs) and their Laman numbers (number of complex relaizations).
Laman graphs and Laman numbers for graphs up to 10 vertices are collected in one zip file.
LamanGraphs<N>.txt/.zip 
	These files contain all Laman graphs with N vertices. The graphs are stored in integer representation. 
	A description on how to read the represnetation can be found below.
Realizations<N>.txt/.zip 
	These files contain the Laman numbers of the Laman graphs in respective order.
	The algorithm that was used to compute these numbers is presented in [1,2].
	An implementaion can be found at [3].
	
Representation of graphs.
We representat a graph by an integer. This integer is obtained from the adjacency matrix.
We take the upper triangle from the matrix and store entries row by row in a list.
This list is interpreted as digits in base two.
The integer is the base 10 representation.
The graph with adjecency matrix
	0 1 1
	1 0 1
	1 1 0
is represented by (111)_2 = 7.
The graph with adjecency matrix
	0 0 1 1
	0 0 1 1
	1 1 0 1
	1 1 1 0
is represented by (011111)_2=31.

In Mathematica the adjecency matrix can be obtained from the integer by the following code:
	G2Mat[g_Integer] := G2Mat[g, Floor[(3 + Sqrt[8*Floor[Log[2, g]] + 1]) / 2]];
	G2Mat[g_Integer, n_Integer] := (# + Transpose[#])&[PadLeft[
		Table[Take[#, {(2n-i)*(i-1)/2+1, (2n-i-1)*i/2}], {i, n}]&[PadLeft[IntegerDigits[g, 2], n*(n-1)/2]], {n, n}]];






References
[1] J. Capco, M. Gallet, G. Grasegger, C. Koutschan, N. Lubbes, and J. Schicho.
	The number of realizations of a Laman graph.
	SIAM Journal on Applied Algebra and Geometry, 2(1):94–125, 2018
	doi: 10.1137/17M1118312

[2] J. Capco, M. Gallet, G. Grasegger, C. Koutschan, N. Lubbes, and J. Schicho.
	Computing the number of realizations of a Laman graph.
	Electronic Notes in Discrete Mathematics, 61:207–213, 2017
	The European Conference on Combinatorics, Graph Theory and Applications (EUROCOMB’17).
	doi: 10.1016/j.endm.2017.06.040

[3] J. Capco, M. Gallet, G. Grasegger, C. Koutschan, N. Lubbes, and J. Schicho
	An algorithm for computing the number of realizations of a Laman graph
	doi: 10.5281/zenodo.1245506


Data manager
Georg Grasegger