Make Only Certain Posts Visible in a Category

Is it possible to make some posts in a category invisible and not others? I know there’s a plugin to make whole categories invisible, but I just need to make a few posts in a category invisible when I’m listing posts in a category.

See, each post’s category has a specific graphic and menu applied to it (the menu is a list of the post titles in that category). I need these invisible posts to have that correct menu and graphic, but not be a part of the menu.

Here’s the code for my menus:

<?php
$cat = get_the_category();
$cat = $cat[0];
$posts = get_posts("category=" . $cat-
>cat_ID . "&numberposts=11&order=ASC");
if( $posts ) :
?>
<ul class="ul1">
<?php foreach( $posts as $post ) :
setup_postdata( $post ); ?>
<li><a href="<?php the_permalink() ?>">
<?php the_title(); ?></a></li>
<?php endforeach; ?>
</ul>
<?php endif; ?>