A {\em fault-tolerant} structure for a network is required to continue
functioning following the failure of some of the network's edges or vertices.
In this paper, we address the problem of designing a {\em fault-tolerant}
additive spanner, namely, a subgraph H of the network G such that
subsequent to the failure of a single vertex, the surviving part of H still
contains an \emph{additive} spanner for (the surviving part of) G, satisfying
dist(s,t,Hβ{v})β€dist(s,t,Gβ{v})+Ξ² for every
s,t,vβV. Recently, the problem of constructing fault-tolerant additive
spanners resilient to the failure of up to f \emph{edges} has been considered
by Braunschvig et. al. The problem of handling \emph{vertex} failures was left
open therein. In this paper we develop new techniques for constructing additive
FT-spanners overcoming the failure of a single vertex in the graph. Our first
result is an FT-spanner with additive stretch 2 and O(n5/3)
edges. Our second result is an FT-spanner with additive stretch 6 and
O(n3/2) edges. The construction algorithm consists of two main
components: (a) constructing an FT-clustering graph and (b) applying a modified
path-buying procedure suitably adopted to failure prone settings. Finally, we
also describe two constructions for {\em fault-tolerant multi-source additive
spanners}, aiming to guarantee a bounded additive stretch following a vertex
failure, for every pair of vertices in SΓV for a given subset of
sources SβV. The additive stretch bounds of our constructions are 4
and 8 (using a different number of edges)