What is the exact algorithm to find maximum clique of a given unit disk graph?
First note that the algorithm is expressed in the so-called proximity model (see preliminaries), so in what follows the points considered will be the centers of your circles, and $d$ will be their common diameter.  

It may be confusing, as the circles drawn in the observations are not directly related to the circles from the intersection model (the radius is $d(a,b)$).



With the notations of the article, $H_{A,B}$ is finite because $V$ itself is finite ($H_{A,B}$ is a subset of the $n$ input points).

Input: d, G = (V, E) with V set of points in the plane, and E is the set
       of {u, v} s.t. d(u, v) &lt;= d
best clique = empty
for all pair of points (u, v) from V:
    let V' = set of vertices w of V s.t. d(u, w) &lt;= d and d(v, w) &lt;= d
    let E' = set of {x, y} s.t. x and y are in V' and d(x, y) &gt; d
    // G' = (V', E') is bipartite
    candidate clique = MaxIndependentSet(G')
    best clique = max(best clique, candidate clique)
return best clique


If you want to use Hopcroft-Karp, you may need to have an explicit partition $V'=A\cup B$ s.t. all edges of $E'$ are connecting a vertex of $A$ and a vertex of $B$. This can be done by checking the sign of the cross product $(\textbf{w}-\textbf{u})\times(\textbf{v}-\textbf{u})$.