The wealth of information contained in online social networks has created a demand for the publication of such data as graphs. Yet, publication, even after identities have been removed, poses a privacy threat. Past research has suggested ways to publish graph data in a way that prevents the re-identification of nodes. However, even when identities are effectively hidden, an adversary may still be able to infer linkage between individuals with sufficiently high confidence. In this paper, we focus on the privacy threat arising from such link disclosure. We suggest L-opacity, a sufficiently strong privacy model that aims to control an adversary’s confidence on short multiedge linkages among nodes. We propose an algorithm with two variant heuristics, featuring a sophisticated look-ahead mechanism, which achieves the desired privacy guarantee after a few graph modifications. We empirically evaluate the performance of our algorithm, measuring the alteration inflicted on graphs and various utility metrics quantifying spectral and structural graph properties, while we also compare them to a recently proposed, albeit limited in generality of scope, alternative. Thereby, we demonstrate that our algorithms are more general, effective, and efficient than the competing technique, while our heuristic that preserves the number of edges in the graph constant fares better overall than one that reduces it.