  We present an adaptive refinement algorithm for T-splines on unstructured 2D
meshes. While for structured 2D meshes, one can refine elements alternatingly
in horizontal and vertical direction, such an approach cannot be generalized
directly to unstructured meshes, where no two unique global mesh directions can
be assigned. To resolve this issue, we introduce the concept of direction
indices, i.e., integers associated to each edge, which are inspired by theory
on higher-dimensional structured T-splines. Together with refinement levels of
edges, these indices essentially drive the refinement scheme. We combine these
ideas with an edge subdivision routine that allows for I-nodes, yielding a very
flexible refinement scheme that nicely distributes the T-nodes, preserving
global linear independence, analysis-suitability (local linear independence)
except in the vicinity of extraordinary nodes, sparsity of the system matrix,
and shape regularity of the mesh elements. Further, we show that the refinement
procedure has linear complexity in the sense of guaranteed upper bounds on a)
the distance between marked and additionally refined elements, and on b) the
ratio of the numbers of generated and marked mesh elements.
