easiest way to disallow ../ in php mkdir

I have a PHP script that creates a folder based on a users name input.

What is the easiest way to deny the user from creating folders up/back in the structure as he could by typing for example "../../folderName" for the folder name?

I could sanitize the users string and remove slashes and/or dots, but is that a 100% safe way of doing it?

        //compile path to create
        $fPath = $path.$parent.'/'.$name;

        //create the folder
        if (!mkdir($fPath)) {
            //setup response json
            $resp = array();
            $resp['success'] = 0;
            $resp['message'] = "Error: Failed to create folder, check permissions on filesystem.";
        }
        else {

            //setup response json
            $resp = array();
            $resp['success'] = 1;
            $resp['message'] = "Folder successfully created in ".$fPath;

        }

        header('Content-Type: application/json');
        echo json_encode($resp);