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An undirected graph $G$ is known to both the prover $P$ and the verifier $V$,but only $P$ knows a subgraph $H$ of $G$. Without revealing any informationabout $H$, $P$ wants to convince $V$ that $H$ is a connected spanning subgraphof $G$, i.e. $H$ is connected and contains all vertices of $G$. In this paper,we propose an unconventional zero-knowledge proof protocol using a physicaldeck of cards, which enables $P$ to physically show that $H$ satisfies thecondition without revealing it. We also show applications of this protocol toverify solutions of three well-known NP-complete problems: the Hamiltoniancycle problem, the maximum leaf spanning tree problem, and a popular logicpuzzle called Bridges.

Physical ZKP for Connected Spanning Subgraph: Applications to Bridges Puzzle and Other Problems