If Apple were to merge with PC World, would this be a vertical or horizontal merger?

Summary: It can't actually occur. There would be vertical and horizontal aspects of the deal. This is very complicated.

Here are the details if you are interested:
PC World, in the UK, I believe is a sub-division of (or sister division with) Curry's and both are divisions in DSG Retail. DSG Retail merged with Carphone Warehouse.
http://www.dixonsretail.com/merger?hash=e185152d519f6f4655963c012e1f7724
All now may be known as Dixon's Carphone Group
http://www.dixonscarphonegroup.com/

The PC World web site, at the bottom of the page says:
About DSG Retail Ltd Corporate site and links to Dixon's Carphone.

PC World USA
http://www.pcworld.com/...world.com/
is IDG Consumer & SMB
PCWorld, Macworld, TechHive and Greenbot content and community.

Now, PC World as an entity is a brand name and not a company for a merger and therefore no merger is possible. That would be the correct answer.

It is hard to envision the purchase of the PC World brand name by Apple as a purchase and not a merger.
In the US, there's Macworld as a publication brand so buying PC World brand does not make sense. In the UK, PC World, if a store title, Apple could purchase the physical stores and brand owned by Dixon's Carphone Group, but then they would be selling their own and competitors products.

It becomes very confusing in every aspect of the question.

There could be a question posed if Apple bought a retailer selling their products and competitors products which goes completely against Apple philosophy it is both vertical as purchasing a retailer of their own products, and horizontal, as buying stores up that Apple owns Apple stores and then they would own other product stores.

A vertical merger is buying suppliers upstream or retailers or servicers downstream.

A horizontal merger is buying a competitor (or combining of equals as a merger) working in the same industry.

If Apple bought HP, Dell, Asus, Acer, Lenovo, etc - Horizontal Merger
If Apple bought Foxconn-HonHai that would be vertical. When Apple bought software and hardware companies used in its products that is vertical.
Apple buys tech companies for use mostly in their products or to close out sources from competitors in an anti-competition move:
https://en.wikipedia.org/...s_by_Apple

They bought Authentec, maker of pc security devices such as Fingerprint readers that were being used in Lenovo Thinkpads in 2012. Cheers dude very helpful

Cheers dude very helpful