"This whole thing began with a question that I asked myself - "how many bosses are reachable right after arriving in Firelink Shrine?".
While it was easy to answer that question from memory, I still decided to search if there is some sort of boss flowchart on the Internet.
All I could find is area progression flowchart and that wasn't what I was looking for, so I decided to make one myself.
This ended up being harder than I initially thought - I probably went through half a dozen different iterations until I found (hopefully) the right order, and twice as many to arrange the flowchart decently.

When multiple arrows lead to one boss it means that every single one of connected bosses has to be defeated before starting that one.

This flowchart doesn't acknowledge any unintended area/boss skips or glitches. This includes but is not limited to: Ceaseless skip, Sen's Gate skip, Firesage Skip and Quelaag Skip.

Obviously Firelink Shrine is not a boss - but I wanted to include it for a few reasons:
-It's the main hub area, and unlike other souls games it's not a warp only nexus - from it you can reach pretty much every area in the game on foot.
-It serves to divide the Northern Undead Asylum (pretty much a tutorial) from the actual game.
-It's a throwback to my original question.

I tested everything in Dark Souls: Prepare to Die Edition, but I don't think anything would change in the Remastered version.
For the whole testing process I used draw.io to arrange and rearrange the flowchart whenever necessary.
Once I finished I exported it to PDF and edited the font, colors and background in Illustrator.

You can download higher resolution versions from my dA: https://www.deviantart.com/patryk567/art/Dark-Souls-boss-order-flowchart-802512790"