Let me explain. Let's say my keyword I want to optimize is "green giraffes". But the domain greengiraffes.com (singular, plural, no hyphen, hyphen, etc.) is not available. I know that the search results for "green giraffes" and "about green giraffes" are essentially the same because "about" is a "stop word". Does that therefore also mean that the domain name "aboutgreengiraffes.com" is as good as "greengiraffes.com" in terms of SEO value? Are all stop words equal in that regard, or a shorter one (such as "e" or "z") is better?
But what's a Google "stop word".
I don't think the modern Google uses stop words anymore in the traditional sense... a common word that is ignored in the search for the sake of efficiency. This I presume is the meaning that you are referring to?
If stop words were still used then (as mentioned in my comment) the two searches you cite, 'green giraffes' and 'about green giraffes' would be identical, but they are not. They are only similar because you are searching for 'green giraffes'!
Take, for example, your two searches on google.com:
Search results for: green giraffes
1. www.green-giraffe.eu/ 2. https://www.gsgatl.org/what-we-do/lgg/pages/default.aspx 3. www.amazon.com › Baby › Diapering › Diaper Pails & Refills 4. [images] 5. gggiraffe.blogspot.com 6. www.gogreengiraffe.com/ 7. www.etsy.com/search?includes%5B%5D=tags...green+giraff 8. greenokapi.tumblr.com/ 9. www.lyricsmania.com/people_who_love_giraffes_lyrics_hank 10. www.hwtm.com/index.cfm?page=albums/view_album... 11. www.newzoo.org/animals/giraffes
Search results for: about green giraffes
- no change,
* new entry,
- 1. www.green-giraffe.eu/ +3 2. gggiraffe.blogspot.com +1 3. [images] * 4. thegreengiraffe.ca/ -3 5. https://www.gsgatl.org/what-we-do/lgg/pages/default.aspx * 6. www.greengiraffephotography.com/ -1 7. www.gogreengiraffe.com/ * 8. www.linkedin.com/company/green-giraffe-capital * 9. www.facebook.com/pages/...Green-Giraffe/14717010530044... -7 10. www.amazon.com › Baby › Diapering › Diaper Pails & Refills -4 11. www.etsy.com/search?includes%5B%5D=tags...green+giraffe
There are 4 new sites in the top 10 for 'about green giraffes' than for 'green giraffes'. If you search on google.co.uk (my native search) then there are 6 new sites!
As understand it, stop words were used by Google some years ago when it used a term based search algorithm. It now uses a phrase based algorithm, where words are used in context. If 'about' is relevant to the search then it will be used.
More information on the changing use of Google stop words:
So, to answer your question...
Does that mean ... the domain name "aboutgreengiraffes.com" is as good as "greengiraffes.com" in terms of SEO value?
It is neither better or worse, it is simply different.
However, if a user was to search for green giraffes, taking into account all the other metrics that Google uses in order to determine relevance then the difference in the domain name between aboutgreengiraffes.com and greengiraffes.com is probably negligible.
It would make no difference because the word about is not between your phrase for example if stop words still exist then greenaboutgiraffes.com would be bad but since the phrase you care about is greengiraffes.com then having aboutgreengiraffes.com is not effecting your two word phrase at all. Your main concern should be the value of keyword being at the beginning, this is said to be better for Google serps for article titles keyword but i am not sure about domain names. for example aboutgreengiraffes.com would be worse than greengiraffesabout.com because it's not at the beginning.