Graphs with flexible labelings allowing injective realizations

Abstract

Given a graph, we ask whether it is possible to find a flexible labeling, namely, edge lengths such that there are infinitely many compatible realizations, modulo rigid motions. Even if a graph is generically rigid, the non-generic edge lengths may still be flexible. In case realizations are not required to be injective, the graphs with a flexible labeling are characterized by the existence of a special edge coloring that is called NAC-coloring. In this talk we focus on graphs and labelings allowing infinitely many injective realizations. We provide a necessary condition on the movability of a graph that is also based on the NAC-colorings of the graph. We show that the condition is also sufficient for graphs with at most 8 vertices, which is not true in general. This is joint work with Georg Grasegger and Josef Schicho.

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Jan Legerský
PhD Student
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