How to publish/un-publish my listing
To publish or un-publish your listing, please click on "My account" in the upper right hand corner and from the drop down menu choose "My Home".
On the resulting page left hand side, you will see the listing menu. Scroll down to the "Home published" option and toggle the button to show "No" for un-published listing or "Yes" for a published listing.
If you decide to un-publish your listing, your listing won't appear in the search results anymore and it won't be visible to other HomeExchange members. This means that you will not receive any new exchange requests until you decide to publish it again, which you can do at any time ;)
How to delete my listing
To delete one of your listings, click on "My account" in the right hand corner. From the drop down menu choose the listing you want to delete. Once on the listing page, on the left hand side, scroll down a bit and click on "I would like to delete my home".
Do note that if you have a pending exchange, you will not be able to delete your listing.