  Let $G = (N,E,w)$ be a weighted communication graph (with weight function $w$
on $E$). For every subset $A \subseteq N$, we delete in the subset $E(A)$ of
edges with ends in $A$, all edges of minimum weight in $E(A)$. Then the
connected components of the corresponding induced subgraph constitute a
partition of $A$ that we call $P_{\min}(A)$. For every game $(N, v)$, we define
the $P_{\min}$-restricted game $(N, \bar{v})$ by $\bar{v}(A) = \sum_{F \in
P_{\min}(A)} v(F)$ for all $A \subseteq N$. We prove that we can decide in
polynomial time if there is inheritance of $\mathcal{F}$-convexity from $(N,
v)$ to the $P_{\min}$-restricted game $(N, \bar{v})$ where
$\mathcal{F}$-convexity is obtained by restricting convexity to connected
subsets.
