Complexity status of restricted k-clique Restricted $k$-clique:

Input: $(G,v,k)$ where $v$ is vertex in $V$

Output: k-clique containing vertex $v$. 

What is the space and time complexity status of this Restricted $k$-clique problem? 

Is it same as $k$-clique problem?

I trying to show this problem is in $L$ or $NL$
Yes. It is essentially same as the Clique problem. Imagine a clique containing $n$ nodes. Your problem is then asking for a Clique containing $n-1$ nodes, such that all of them are adjacent to vertex $v$. $v$ is connected to all vertices in the graph. The problem is still NPComplete.