# Karp hardness of a supportive clique in digraphs

In a directed graph (digraph) $$G(V,A)$$, a set of vertices $$C\subseteq V$$ is called a supportive clique if (1) for every $$u\neq v$$ in $$C$$, either $$(u,v)\in A$$ or $$(v,u)\in A$$ and (2) for every $$u\in C$$, there exists $$v,w\in C$$ such that $$(u,v), (w,v)\in C$$.

In other words, a supportive clique in digraph is a standard clique if we omit the direction of all arcs, when taking back into account the direction of arcs, every vertex in a supportive have at least an incoming arc from and an outgoing to other vertices in the clique.

SUPPORTIVE CLIQUE:

Input: A digraph $$G(V,A)$$ and $$k\in\mathbb{N}$$

Output: YES if there exists a supportive clique of size $$k$$ in $$G$$, otherwise NO

What is the complexity of this problem?