  An alternative foundation for 2-categories is explored by studying
graph-theoretically a partial operation on 2-cells named juncture, which can
replace vertical and horizontal composition. Juncture is a generalized vertical
composition of 2-cells that need not involve the whole target and the whole
source; it may involve them only partly, provided the result is again a 2-cell.
Since commuting diagrams of arrows of ordinary categories may be conceived as
invertible 2-cells, this study concerns ordinary category theory too. The
operation of juncture has a connection with proof theory, where it corresponds
to a kind of cut rule on sequents, and it is related also to an operation on
which the notion of operad can be based. The main achievement of the work is a
detailed description of the specific planarity involved in juncture and graphs
of 2-cells, comparable to the usual combinatorial characterizations of
planarity in graph theory. This work points out to an alternative foundation
for bicategories, i.e. weak 2-categories, and more generally weak n-categories.
