Help with Wordpress Walker Function [duplicate]

Question

I need some help customizing a walker function. I don't fully undrestand it, but it's helped me get half of what I need.

class themeslug_walker_nav_menu extends Walker_Nav_Menu {


    // add classes to ul sub-menus
    function start_lvl( &$output, $depth ) {
        // depth dependent classes
        $indent = ( $depth > 0  ? str_repeat( "\t", $depth ) : '' ); // code indent
        $display_depth = ( $depth + 1); // because it counts the first submenu as 0
        $classes = array(
            ' dropdown',
            /*?>( $display_depth % 4  ? ' ' : '' ),
            ( $display_depth >=4 ? '' : '' ),
            '-' . $display_depth<?php */
            );
        $class_names = implode( ' ', $classes );

        // build html
        $output .= "\n" . $indent . '<ul class="' . $class_names . '">' . "\n";
    }

}

Here is what I have so far, and it works great to add a submenu to the ul at the 2nd and third levels. I need to figure out how to add a custom class to the parent ul. I can do it in query but it's not quite the solution I would like. Any help is appreciated.