Healing algorithms play a crucial part in distributed P2P networks where failures occur continuously and frequently. Several self-healing algorithms have been suggested recently [IPDPS'08, PODC'08, PODC'09, PODC'11] in a line of work that has yielded gradual improvements in the properties ensured on the graph. This work motivates a strong general phenomenon of edge-preserving healing that aims at obtaining self-healing algorithms with the constraint that all original edges in the graph (not deleted by the adversary), be retained in every intermediate graph. The previous algorithms, in their nascent form, are not explicitly edge preserving. In this paper, we show they can be suitably modified (We introduce Xheal+, an edge-preserving version of Xheal[PODC'11]). Towards this end, we present a general self-healing model that unifies the previous models. The main contribution of this paper is not in the technical complexity, rather in the simplicity with which the edge-preserving property can be ensured and the message that this is a crucial property with several benefits. In particular, we highlight this by showing that, almost as an immediate corollary, subgraph densities are preserved or increased. Maintaining density is a notion motivated by the fact that in certain distributed networks, certain nodes may require and initially have a larger number of inter-connections. It is vital that a healing algorithm, even amidst failures, respect these requirements. Our suggested modifications yield such subgraph density preservation as a by product. In addition, edge preservation helps maintain any subgraph induced property that is monotonic. Also, algorithms that are edge-preserving require minimal alteration of edges which can be an expensive cost in healing - something that has not been modeled in any of the past work.