| Displaying 1-7 of 7 results found.
page
1
Number of minimally rigid graphs with n vertices constructible by Henneberg type I moves.
+10
6
1, 1, 1, 1, 3, 11, 61, 499, 5500, 75635, 1237670, 23352425, 498028767, 11836515526, 310152665647
COMMENTS
A graph is called rigid if, when we fix the length of each edge, it has only finitely many embeddings in the plane. A graph is called minimally rigid (or a Laman graph) if there is no edge that can be omitted while keeping the rigidity property. Laman graphs can be constructed by applying successively Henneberg moves (of type I or type II), starting with the graph that consists of two vertices joined by an edge. Here we consider Laman graphs that can be obtained by using only Henneberg type I moves, which means: adding one vertex and joining it with two different existing vertices.
EXAMPLE
A single vertex is rigid.
The graph consisting of two vertices joined by an edge is rigid.
A triangle is rigid and it is obtained by a single Henneberg type I move.
One more such move yields the only Laman graph with four vertices.
Also all three Laman graphs with five vertices can be constructed with type I moves. Therefore the first five entries of this sequence agree with A227117. An example of a Laman graph that cannot be constructed using only Henneberg type I moves is the complete bipartite graph K(3,3).
MATHEMATICA
Table[Length[H1LamanGraphs[n]], {n, 3, 7}] (* see link *)
PROG
(Nauty with Laman plugin) gensparseg $n -H -u #see link
Number of bipartite Laman graphs on n vertices.
+10
3
1, 1, 0, 0, 0, 1, 1, 5, 19, 123, 871, 8304, 92539, 1210044, 17860267, 293210063, 5277557739
COMMENTS
All the Laman graphs (in other words, minimally rigid graphs) can be constructed by the inductive Henneberg construction, i.e., a sequence of Henneberg steps starting from K_2. A new vertex added by a Henneberg move is connected with two or three of the previously existing vertices. Hence, the chromatic number of a Laman graph can be 2, 3 or 4. One can hypothesize that the set of 3-chromatic Laman graphs is the largest and that bipartite graphs are relatively rare. The first nontrivial bipartite Laman graph is K_{3,3}. An infinite sequence of such graphs can be obtained from K_{3,3} by Henneberg moves of the first type (i.e., adding a vertex and connecting it with two of the existing vertices from the one part).
LINKS
F. Hüffner, tinygraph, software for generating integer sequences based on graph properties. Christoph Koutschan, Mathematica program for generating a list of non-isomorphic Laman graphs on n vertices.
MATHEMATICA
Table[Length[
Select[LamanGraphs[n],
BipartiteGraphQ[AdjacencyGraph[G2Mat[#]]] &]], {n, 6, 9}] (* using the program by Christoph Koutschan for generating Laman graphs, see A227117 *)
EXTENSIONS
a(13)-a(15) added using tinygraph by Falk Hüffner, Oct 20 2019
Number of unlabeled minimally rigid graphs in 3D on n vertices.
+10
3
1, 1, 1, 4, 26, 374, 11487, 612884, 48176183, 5115840190, 698180921122
COMMENTS
All minimally rigid graphs in 3D on n vertices can be constructed from the minimally rigid graphs in 3D on n-1 vertices by use of three types of constructions called the Henneberg constructions. In the first type a new vertex is added to the graph and three new edges are added connecting the new vertex to three different vertices which were already part of the graph. In the second type of construction, two adjacent vertices, say v_1 and v_2, are selected. The edge between v_1 and v_2 is deleted. A new vertex w is added to the graph, as well as the edges (v_1,w), (v_2,w), (v_3,w), and (v_4,w), where v_3 and v_4 are other vertices of the graph. The third construction chooses two pairs of adjacent vertices v_1,v_2 and v_3,v_4, where v_3 might be equal to v_2. The edges (v_1,v_2) and (v_3,v_4) are deleted. A new vertex w is added to the graph. If v_2!=v_3, the edges (v_1,w), (v_2,w), (v_3,w), (v_4,w), and (v_5,w) are added, where v_5 is another vertex of the graph. If v_2=v_3, other two vertices v_5,v_6 are chosen and the edges (v_1,w), (v_2,w),(v_4,w), (v_5,w), and (v_6,w) are added.
The first two constructions preserve rigidity whereas the third does not necessarily preserve rigidity. Nevertheless the third construction is needed to get all minimally rigid graphs in 3D. Up to 11 vertices the first two constructions suffice.
Each of these three constructions adds one to the number of vertices and three to the number of edges, i.e., all those graphs have 3n-6 edges for n vertices. Not all graphs with that number of edges are minimally rigid in 3D.
MATHEMATICA
Table[Length[H12GeiringerGraphs[n]], {n, 4, 11}] (* see Link *)
CROSSREFS
Cf. A227117 (number of minimally rigid graphs in 2D on n vertices). Cf. A374745 (number of (3,6)-tight graphs).
Number of 4-chromatic Laman graphs on n vertices.
+10
2
1, 8, 102, 1601, 29811, 636686, 15298955, 407748141, 11932078866
COMMENTS
All the Laman graphs (in other words, minimally rigid graphs) can be constructed by the inductive Henneberg construction, i.e., a sequence of Henneberg steps starting from K_2. A new vertex added by a Henneberg move is connected with two or three of the previously existing vertices. Hence, the chromatic number of a Laman graph can be 2, 3 or 4. One can hypothesize that the set of 3-chromatic Laman graphs is the largest and that bipartite graphs are relatively rare. The simplest example of a 4-chromatic Laman graph is the Moser spindle.
LINKS
Christoph Koutschan, Mathematica program for generating a list of non-isomorphic Laman graphs on n vertices. Eric Weisstein's World of Mathematics, Moser spindle is a 4-chromatic Laman graph.
MATHEMATICA
Table[Length[
Select[LamanGraphs[n],
IGChromaticNumber[AdjacencyGraph[G2Mat[#]]] == 4 &]], {n, 7, 9}]
(* using the program by Christoph Koutschan for generating Laman graphs, see A227117, and IGraph/M interface by Szabolcs Horvát *)
PROG
(nauty with Laman plugin) gensparseg $n -K2 | countg --N # see link
Maximal Laman number among all minimally rigid graphs on n vertices.
+10
1
1, 1, 2, 4, 8, 24, 56, 136, 344, 880, 2288, 6180, 15536, 42780
COMMENTS
The Laman number gives the number of (complex) embeddings of a minimally rigid graph in 2D, modulo translations and rotations, when the edge lengths of the graph are chosen generically. In general, this number is larger than the number of real embeddings. Equivalently, the Laman number of a graph is the number of complex solutions of the quadratic polynomial system {x_1 = y_1 = x_2 = 0, y_2 = l(1,2), (x_i - x_j)^2 + (y_i - y_j)^2 = l(i,j)^2}, for all (i,j) such that the vertices i and j are connected by an edge (w.l.o.g. we assume that there is an edge between the vertices 1 and 2). The quantities l(i,j) correspond to the prescribed edge "lengths" (they can also be complex numbers).
A graph that is constructed only by Henneberg moves of type 1 (i.e., adding one new vertex and connecting it with two existing vertices), has Laman number 2^(n-2). The smallest minimally rigid graph that cannot be constructed in this way, is the 3-prism graph with 6 vertices. Therefore the sequence grows faster than 2^(n-2) for n >= 6.
We know that a graph with n <= 13 vertices achieving the maximal Laman number is unique. We do not know if this is necessarily true for more vertices.
REFERENCES
J. Capco, M. Gallet, G. Grasegger, C. Koutschan, N. Lubbes, J. Schicho, The number of realizations of a Laman graph, SIAM Journal on Applied Algebra and Geometry 2(1), pp. 94-125, 2018.
I. Z. Emiris, E. P. Tsigaridas, A. E. Varvitsiotis, Algebraic methods for counting Euclidean embeddings of graphs. Graph Drawing: 17th International Symposium, pp. 195-200, 2009.
G. Grasegger, C. Koutschan, E. Tsigaridas, Lower bounds on the number of realizations of rigid graphs, Experimental Mathematics, 2018 (doi: 10.1080/10586458.2018.1437851).
EXAMPLE
A graph with one vertex can be drawn in the plane in a unique way, and similarly the graph with two vertices connected by an edge. The unique minimally rigid graph with three vertices is the triangle, which admits two different embeddings (they differ by reflection). The unique minimally rigid graph with four vertices is a quadrilateral with one diagonal (i.e., we have five edges). By fixing the diagonal, each of the two triangles can be flipped independently, yielding four different embeddings.
PROG
(nauty) # See nauty plugin in Links.
EXTENSIONS
a(14) computed and added by Jose Capco, Oct 02 2023
Number of (3/2,2)-tight graphs with 2n vertices, or kinematic chains with 2n links.
+10
0
1, 1, 2, 16, 230, 6856, 318162, 19819281
COMMENTS
A 2n-vertex graph is (3/2,2)-sparse if every subgraph with k vertices has at most (3/2)k-2 edges, and (3/2,2)-tight if in addition it has exactly 3n-2 edges; see Lee and Streinu (2008). These graphs represent two-dimensional mechanical systems formed by 2n rigid bodies (links), connected at joints where exactly two links are pinned together and can rotate relative to each other, with the entire system having one degree of freedom and having no rigid subsystems. The vertices of the graph represent links and the edges represent joints.
EXAMPLE
For n=1 the single example (a graph with two vertices and one edge) is represented by familiar mechanical systems including door hinges and pairs of scissors. For n=3 the a(3)=2 solutions are the 6-vertex 7-edge graphs Theta(1,3,3) and Theta(2,2,3), each of which has two degree-three vertices connected by three paths of the given lengths. These correspond respectively to the Watt linkage (two four-bar linkages sharing a pair of adjacent links) and the Stephenson linkage.
Number of unlabeled Laman graphs on n vertices of degree at most 4.
+10
0
1, 1, 1, 1, 3, 10, 37, 189, 1145, 8089, 64683, 571949, 5499343, 56899844, 628729114, 7380050235
PROG
(nauty with Laman plugin) gensparseg $n -D4 -K2 -u # see link
Search completed in 0.020 seconds
| 