Writing JSON object to .json file on server

You are double-encoding. There is no need to encode in JS and PHP, just do it on one side, and just do it once.

// step 1: build data structure
var data = {
    metros: graph.getVerticies(),
    routes: graph.getEdges()
}

// step 2: convert data structure to JSON
$.ajax({
    type : "POST",
    url : "json.php",
    data : {
        json : JSON.stringify(data)
    }
});

Note that the dataType parameter denotes the expected response type, not the the type you send the data as. Post requests will be sent as application/x-www-form-urlencoded by default.

I don’t think you need that parameter at all. You could trim that down to:

$.post("json.php", {json : JSON.stringify(data)});

Then (in PHP) do:

<?php
   $json = $_POST['json'];

   /* sanity check */
   if (json_decode($json) != null)
   {
     $file = fopen('new_map_data.json','w+');
     fwrite($file, $json);
     fclose($file);
   }
   else
   {
     // user has posted invalid JSON, handle the error 
   }
?>