  In card-based cryptography, a deck of physical cards is used to achieve
secure computation. A shuffle, which randomly permutes a card-sequence along
with some probability distribution, ensures the security of a card-based
protocol. The authors proposed a new class of shuffles called graph shuffles,
which randomly permutes a card-sequence by an automorphism of a directed graph
(New Generation Computing 2022). For a directed graph $G$ with $n$ vertices and
$m$ edges, such a shuffle could be implemented with pile-scramble shuffles with
$2(n+m)$ cards. In this paper, we study graph shuffles and give an
implementation, an application, and a slight generalization of them. First, we
propose a new protocol for graph shuffles with $2n+m$ cards. Second, as a new
application of graph shuffles, we show that any cyclic group shuffle, which is
a shuffle over a cyclic group, is a graph shuffle associated with some graph.
Third, we define a hypergraph shuffle, which is a shuffle by an automorphism of
a hypergraph, and show that any hypergraph shuffle can also be implemented with
pile-scramble shuffles.
