Firefox's tabbed browsing feature encourages breadth-first search. If I click on a link the tab containing the page linked to appears as the rightmost tab, and I generally work through tabs from left to right. Breadth-first search can be implemented in this way -- we maintain a list of pages to be looked at, and the first page to enter the queue is also the first page to leave it.
Depth-first browsing wouldn't be too much different on a cosmetic level -- it could be set up by having the new tab appear immediately after the current tab, giving a stack of pages to view instead of a queue. I suspect the subjective experience of Internet browsing would feel much different from such a point of view -- browsing often seems to lead to shallow knowledge. (If one had time to search the entire Internet breadth-first search and depth-first search would eventually visit the same set of pages -- but who has that kind of time?) The "optimal" algorithm for finding particular information isn't strictly breadth-first or depth-first, though; if you think about how you search when you look for a specific piece of information, you don't routinely follow the leftmost tab or the rightmost tab, but instead click on whatever tab subjectively seems like it would give the best information.
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This is the sort of thing that atches me out whenever I'm using IE 7, being used to the Firefox/Safari method. New tabs appear just to the right of the current tab or the last tab that was opened in the background. I think that's it, no WinPC near-by to check on.
If you'd like to get the other behavior in Firefox, there is (naturally) a plugin: Tabs Open Relative.
I guess you could say I use a hybrid approach of depth-first and breadth-first browsing. I'll open tabs when skimming material (i.e., from Bloglines or reddit) but if I hit something that looks worthy of some more attention I'll open it in a new window and switch to it, possibly opening new tabs as I browse that topic.
The other mode of search that gets interspersed in all of this it hitting Google or Wikipedia for background information. In terms of graph search that's like leaping from node to node without any regard for edges.
My new tabs open next to my current tab. The tablist is sort of like a ToDo/ToRead list for me.
I usually have multiple windows open. One for my feed reader and one for mail. Tabs for other small tasks are just opened in one of these. Or I might open a new window if I need many tabs for a task.
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