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Let $G=(V,E)$ be any undirected graph on $V$ vertices and $E$ edges. A path$\textbf{P}$ between any two vertices $u,v\in V$ is said to be $t$-approximateshortest path if its length is at most $t$ times the length of the shortestpath between $u$ and $v$. We consider the problem of building a compact datastructure for a given graph $G$ which is capable of answering the followingquery for any $u,v,z\in V$ and $t>1$:  Report $t$-approximate shortest path between $u$ and $v$ when vertex $z$fails  We present data structures for the single source as well all-pairs versionsof this problem. Our data structures guarantee optimal query time. Mostimpressive feature of our data structures is that their size {\em nearly} matchthe size of their best static counterparts.

Approximate Shortest Paths Avoiding a Failed Vertex: Optimal Size Data Structures for Unweighted Graph