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Highlights • Redefine the verification condition in (k, n)-MCPVCS. • Define a minimum (k, n)-connected graph problem. • Propose a (k , n)-MCPVCS with large size of verification image based on minimum (k, n)-connected graph. Recently, Lin et al. [29] introduced a (k , n) multi-factor cheating prevention visual cryptographic scheme (MCPVCS) by hiding verification images into shadow images. Through authenticating the verification image and the reconstructed image simultaneously, Lin et al.’s MCPVCS can prevent cheating attack that dishonest participants may collude and cheat the honest one to get a wrong secret image. However, the large n reduces the size of verification image and makes the verification difficult. In this paper, a (k , n)-MCPVCS with the large size of verification image is proposed. There are three main contributions for the proposed (k , n)-MCPVCS: (i) the verification condition in (k , n)-MCPVCS is redefined, (ii) a new minimum (k , n)-connected graph problem is introduced, on which the minimum number of verification images is determined, and (iii) the proposed (k , n)-MCPVCS with the large size of verification image can more easily authenticate the unambiguity of verification image. [ABSTRACT FROM AUTHOR]