GtkWindowiconify

Asks to iconify (i.e. minimize) the specified window. Note that you shouldn’t assume the window is definitely iconified afterward, because other entities (e.g. the user or [window manager][gtk-X11-arch]) could deiconify it again, or there may not be a window manager in which case iconification isn’t possible, etc. But normally the window will end up iconified. Just don’t write code that crashes if not.

It’s permitted to call this function before showing a window, in which case the window will be iconified before it ever appears onscreen.

You can track iconification via the “window-state-event” signal on GtkWidget.