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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 $NL$

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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.

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  • $\begingroup$ If we solve Restricted $k$-clique problem then we can solve $k$-clique problem. But not other way round. $\endgroup$
    – GOLD
    Jun 6, 2017 at 13:03
  • $\begingroup$ The much easier problem of approximating Clique itself is NPComplete ( groups.csail.mit.edu/cis/pubs/shafi/1991-focs-fglss.pdf). And if we consider the largest clique (of size $n$), finding the clique of size $n-1$ is obviously NPComplete. $\endgroup$
    – TheoryQuest1
    Jun 6, 2017 at 13:08
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    $\begingroup$ What @TheoryQuest1 tries to explain I think is that a $k$-clique in a graph $G$ is exactly a restricted $k+1$-clique containing the vertex $v'$ in the graph $G'$ where $G'$ is obtained from $G$ by adding a fresh vertex $v'$ connected to all vertices of $G$. $\endgroup$
    – holf
    Jun 6, 2017 at 13:16
  • $\begingroup$ I should have put it that way. Much more accessible. $\endgroup$
    – TheoryQuest1
    Jun 6, 2017 at 13:18

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