  We study four-dimensional N=1 gauge theories which arise from D3-brane probes
of toric Calabi-Yau threefolds. There are some standing paradoxes in the
literature regarding relations among (p,q)-webs, toric diagrams and various
phases of the gauge theories, we resolve them by proposing and carefully
distinguishing between two kinds of (p,q)-webs: toric and quiver (p,q)-webs.
The former has a one to one correspondence with the toric diagram while the
latter can correspond to multiple gauge theories. The key reason for this
ambiguity is that a given quiver (p,q)-web can not capture non-chiral matter
fields in the gauge theory. To support our claim we analyse families of
theories emerging from partial resolution of Abelian orbifolds using the
Inverse Algorithm of hep-th/0003085 as well as (p,q)-web techniques. We present
complex inter-relations among these theories by Higgsing, blowups and brane
splittings. We also point out subtleties involved in the ordering of legs in
the (p,q) diagram.
