A　nonlinear　vertex-centered　finite　volume　scheme　that　preserves　the　discrete　maximum　principle　(DMP)　is　proposed　for　anisotropic　diffusion　equations　on　polygonal　meshes.　The　new　scheme　is　constructed　from　a　vertex-centered　linearity-preserving　scheme　combined　with　a　nonlinear　correction　technique.　The　key　ingredient　is　the　introduction　of　the　correction　coefficient　and　the　limiter.　Particularly,　the　choice　of　the　limiter　is　discussed　and　two　new　and　effective　limiters　are　proposed.　Different　from　most　existing　nonlinear　DMP-preserving　schemes,　the　new　nonlinear　scheme　does　not　rely　on　the　nonlinear　combination　of　linear　flux　approximations　and　requires　no　interpolation　of　the　auxiliary　unknowns.　More　important　is　that,　it　can　be　proved　theoretically　that　the　new　scheme　is　symmetric　and　coercive　on　general　polygonal　meshes　with　star-shaped　cells.　Moreover,　other　theoretical　results　such　as　the　DMP　property　and　stability　of　the　correction　scheme　are　also　presented.　Numerical　experiments　demonstrate　the　accuracy,　efficiency,　and　DMP　property　of　the　new　scheme　on　general　polygonal　meshes　with　heterogeneous　and　anisotropic　diffusion　tensors.　©　2024　IMACS