Algorithms for Graph Rigidity and Scene Analysis

We investigate algorithmic questions and structural problems concerning graph families defined by ‘edge-counts’. Motivated by recent developments in the unique realization problem of graphs, we give an efficient algorithm to compute the rigid, redundantly rigid, M-connected, and globally rigid components of a graph. Our algorithm is based on (and also extends and simplifies) the idea of Hendrickson and Jacobs, as it uses orientations as the main algorithmic tool. We also consider families of bipartite graphs which occur in parallel drawings and scene analysis. We verify a conjecture of Whiteley by showing that 2d-connected bipartite graphs are d-tight. We give a new algorithm for finding a maximal d-sharp subgraph. We also answer a question of Imai and show that finding a maximum size d-sharp subgraph is NP-hard.