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

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