Concept

Laman graph

In graph theory, the Laman graphs are a family of sparse graphs describing the minimally rigid systems of rods and joints in the plane. Formally, a Laman graph is a graph on n vertices such that, for all k, every k-vertex subgraph has at most 2k − 3 edges, and such that the whole graph has exactly 2n − 3 edges. Laman graphs are named after Gerard Laman, of the University of Amsterdam, who in 1970 used them to characterize rigid planar structures. This characterization, however, had already been discovered in 1927 by Hilda Geiringer. Laman graphs arise in rigidity theory: if one places the vertices of a Laman graph in the Euclidean plane, in general position, there will in general be no simultaneous continuous motion of all the points, other than Euclidean congruences, that preserves the lengths of all the graph edges. A graph is rigid in this sense if and only if it has a Laman subgraph that spans all of its vertices. Thus, the Laman graphs are exactly the minimally rigid graphs, and they form the bases of the two-dimensional rigidity matroids. If n points in the plane are given, then there are 2n degrees of freedom in their placement (each point has two independent coordinates), but a rigid graph has only three degrees of freedom (the position of a single one of its vertices and the rotation of the remaining graph around that vertex). Intuitively, adding an edge of fixed length to a graph reduces its number of degrees of freedom by one, so the 2n − 3 edges in a Laman graph reduce the 2n degrees of freedom of the initial point placement to the three degrees of freedom of a rigid graph. However, not every graph with 2n − 3 edges is rigid; the condition in the definition of a Laman graph that no subgraph can have too many edges ensures that each edge contributes to reducing the overall number of degrees of freedom, and is not wasted within a subgraph that is already itself rigid due to its other edges.

About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Graph Chatbot

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.