Some Irreducible Laman Graphs

Adel Moussaoui (1) , Samy Ait-Aoudia (1)

1 ESI - École Nationale Supérieure d'Informatique [Alger]

In graph theory, Laman graphs [1] describe the minimally rigid plane structures composed of bars and joints. Removing a bar leads to the non-rigidity of the structure. In the domain of geometric constraints solving, this property is commonly referred to as well-constrained graphs or structures. Formally, a graph G=(V, E) where |V|=n and |E|=m is a Laman graph if and only if m=2*n-3 and m’ 2*n’-3 for any induced sub-graph G'=(V',E ') where |V'|=n' and |E'|=m'. Another characterization of rigid structures was given in 1911 by Henneberg [2] for which any minimal rigid plane structure is obtained starting from an edge joining two vertices and adding one vertex at a time using one of the two following operations. Operation HI: add a new vertex v to G, then connect v to two chosen vertices u and w from G via two new edges (v, u) and (v, w). Operation HII: add a new vertex v to G, chose an edge (u, w) and a vertex z to G, then add three edges (v, u), (v, w) and (v, z) to G, finally delete the edge (u, w). An irreducible Laman graph is a graph G=(V, E) where |V|=n, |E|=m , m=2*n-3 and m’<2*n’-3 for any induced sub-graph G'=(V',E ') where |V'| = n' and |E'|=m'. An irreducible Laman graph is a graph that contains no Laman subgraphs. Based on the constraint system reduction algorithm [3], Laman irreducible graphs are obtained using the generator of 2D Geometric Constraint Graphs described in [4]. Laman irreducible graphs with up to 100 vertices are given in the appendix.

Mots clés: Laman Graphs; Henneberg Construction; Irreducible Graphs

Domaines: Géométrie algorithmique [cs.CG]

Adel Moussaoui, Samy Ait-Aoudia. Some Irreducible Laman Graphs. [Research Report] ESI - Ecole nationale Supérieure en Informatique - Alger. 2016. ⟨hal-01521779v2⟩