  Let us consider the local specification system of Gibbs point process with
inhib ition pairwise interaction acting on some Delaunay subgraph specifically
not con taining the edges of Delaunay triangles with circumscribed circle of
radius grea ter than some fixed positive real value $R$. Even if we think that
there exists at least a stationary Gibbs state associated to such system, we do
not know yet how to prove it mainly due to some uncontrolled "negative"
contribution in the expression of the local energy needed to insert any number
of points in some large enough empty region of the space. This is solved by
introducing some subgraph, called the $R$-local Delaunay graph, which is a
slight but tailored modification of the previous one. This kind of model does
not inherit the local stability property but satisfies s ome new extension
called $R$-local stability. This weakened property combined with the local
property provides the existence o f Gibbs state.
