The article seems really pretty clear about that line: market dominance. As another commenter said, it's an extremely well-written and clear article.
Did your price comparison site at any time seem likely to be able to dominate the entire EU market? (you can find the exact criteria on the EU site)
If no, you got no problem to worry about.
If yes, that's a kind of nice problem to have I think.
> What's next? Having to give equal ranking to Yahoo! search results in their search page?
Why would anything be next? We have this rule, it works well, has worked in the past, what's the incentive to make it worse?
Your Yahoo!Search example doesn't make sense to me. Not in context of this ruling, any way. How would Google abuse their market dominance in order to ... what exactly, because you didn't specify any wrongdoing, just a crazy sanction pulled out of thin air.
It's not the market dominance itself that is illegal. It's just that if you do it, dominate a market to such an extent, you are held to special responsibilities not to abuse that dominance. Which totally makes sense.
From the article:
"""Market dominance is, as such, not illegal under EU antitrust rules. However, dominant companies have a special responsibility not to abuse their powerful market position by restricting competition, either in the market where they are dominant or in separate markets."""
It's just like that famous Spiderman quote.
Also the ruling is not about a particular party (like Yahoo in yours) that got wronged by an illegal abuse of Google's market dominance, it's about all of them, about the responsibility not to actively stifle competition. You can still be better than the competition, Google is obviously allowed to "hurt" Yahoo by providing better search results, that's what it means when they say dominance is as such not illegal under antitrust. It's just that, in some (but not all) cases of market dominance, it comes with additional power by virtue of dominance (not product), that allows a company to stifle competition (not because they're better but because the competition is not dominant), and antitrust law states that companies in such a position have a responsibility not to do this. Yet Google did. I'm not really seeing how Google could do something similar that involves Yahoo in your example.
About your One Box (that's the Google instant result thing yes?) example, if Google were in the theatre business, I'm pretty sure the Commission would investigate that case for antitrust as well. How that ruling would go, will depend on the specifics of that hypothetical situation, and whether it qualifies along the guidelines as stated partially in the article, and fully in EU regulations on antitrust (which you can click to for yourself).