Visual Cryptography Scheme for the Mindom Access Structure of a Graph

Let \( G = (V, E) \) be a connected graph with domination number \( \gamma \geq 2 \). In this paper, we discuss the construction of a visual cryptography scheme for the mindom access structure \( \Gamma_D(G) \) with a basis consisting of all \( \gamma \)-sets of \( G \). We prove that the access structure \( \Gamma_D(G) \) is a \( (2, n) \)-threshold access structure if and only if \( n \) is even and \( G = K_n – M \), where \( M \) is a perfect matching in \( K_n \). Further, the \( (k, n) \)-VCS with \( k < n \) can be realized as a \( \Gamma_D(G) \)-VCS if and only if \( k = 2 \) and \( n \) is even. We also construct \( \Gamma_D(G) \)-VCS for several classes of graphs such as complete bipartite graphs, cycles \( C_n \), and \( K_n – C_n \), and we have achieved substantial reduction in the pixel expansion when compared to the VCS constructed by using other known methods.