Locally refined B-splines, which were introduced by Dokken et al. [2], provide a generalization of tensor-
product B-splines to the case of locally refined meshes. While refinement strategies that ensure linear inde-
pendence have been studied recently [4, 3], the construction of LR B-splines may potentially generate basis
functions that possess the even stronger property of local linear independence (LLI). More precisely, LLI en-
sures that exactly (p + 1)^d basis functions take non-zero values on any cell of the mesh and entails optimal
sparsity properties of the matrices that arise, e.g., in applications to numerical simulation. Motivated by the
notion of semi-regular tensor-product B-splines, which was introduced by Weller and Hagen [1], we investigate
related refinement strategies for locally linear independent LR B-splines in the bivariate case.