how to get dependencies from package name?

edit

The output of yum deplist looks like this:

$ yum deplist gtkd
  ...
  dependency: librt.so.1()(64bit)
   provider: glibc.x86_64 2.18-11.fc20
  dependency: mesa-libGL
   provider: mesa-libGL.x86_64 9.2-1.20130919.fc20
   provider: mesa-libGL.i686 9.2-1.20130919.fc20
  dependency: mesa-libGLU
   provider: mesa-libGLU.x86_64 9.0.0-3.fc20
   provider: mesa-libGLU.i686 9.0.0-3.fc20
   ...

Since the dependency line can give a library and not a package name, I would only keep the provider lines with grep provider.

$ yum deplist gtkd | grep provider
   ...
   provider: glibc.x86_64 2.18-11.fc20
   provider: mesa-libGL.x86_64 9.2-1.20130919.fc20
   provider: mesa-libGL.i686 9.2-1.20130919.fc20
   provider: mesa-libGLU.x86_64 9.0.0-3.fc20
   provider: mesa-libGLU.i686 9.0.0-3.fc20
   ...

Then I remove the string "provider:", the extra spaces before and after, and I keep the word just following using sed "s/[ ]*provider:[ ]*\([^ ]*\).*/\1/" for example:

$ yum deplist gtkd | grep provider | sed "s/[ ]*provider:[ ]*\([^ ]*\).*/\1/"
...
glibc.x86_64
mesa-libGL.x86_64
mesa-libGL.i686
mesa-libGLU.x86
mesa-libGLU.i686
...

I can now keep everything before the last period with sed "s/\(.*\)\.[^\.]*/\1/":

$ yum deplist gtkd | grep provider | sed "s/[ ]*provider:[ ]*\([^ ]*\).*/\1/" | sed "s/\(.*\)\.[^\.]*/\1/"
...
glibc
mesa-libGL
mesa-libGL
mesa-libGLU
mesa-libGLU
...

And I sort as you do removing the duplicates with sort -u:

$ yum deplist gtkd | grep provider | sed "s/[ ]*provider:[ ]*\([^ ]*\).*/\1/" | sed "s/\(.*\)\.[^\.]*/\1/" | sort -u
atk
cairo
gdk-pixbuf2
glibc
gstreamer
gstreamer-plugins-base
gtk2
gtksourceview2
libglade2
mesa-libGL
mesa-libGLU
pango

Edit: I have updated the answer according to @marcindulak remark, so it can now work with dependencies whose name contains a dot:

$ yum deplist icedtea-web | grep provider | sed "s/[ ]*provider:[ ]*\([^ ]*\).*/\1/" | sed "s/\(.*\)\.[^\.]*/\1/" | sort -u
bash
chkconfig
glib2
glibc
java-1.7.0-openjdk
libgcc
libstdc++
mozilla-filesystem