Showing/Hiding children of a Structure based on categories
Something like this should do the trick! You want to get all the section parent entries, then search for child entries that are related to the category you want. 

{% set parents = craft.entries.section('your_structure_handle').level(1) %}

{% for parent in parents %}
    {% set children = craft.entries.decendantOf(parent).relatedTo(category) %}
    {% if children|length %}
        &lt;h1&gt;{{parent.title}}&lt;/h1&gt;
        &lt;ul&gt;
        {% for child in children %}
            &lt;li&gt;{{child.title}}&lt;/li&gt;
        {% endfor %}
        &lt;/ul&gt;
    {% endif %}
{% endfor %}