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you’re using a distro with systemd and, as such, it’s a simple
(and universal) command to find out which display manager you’re
using. The following command will give you the name and a bunch
of other information about your display-manager:
systemctl status display-manager
Debian/Ubuntu/etc: cat /etc/X11/default-display-manager
SEL/OpenSUSE/etc: cat /etc/sysconfig/displaymanager
RHEL/Fedora/etc: cat /etc/sysconfig/desktop