The eastern and western parts of the Roman Empire were two parts
of a sole Roman Empire and never attacked each other. They did not
attack each other because they were part of the same empire. In
fact there ware alternations of periods with co-emperors (with one
in charge of the east and one in charge of the west) and periods
with a sole emperor for the whole of the empire.
In 307 Galerius, the senior emperor in the eastern part,
attacked Maxentius, a usurper emperor in Italy. However, this was
not an attack of the east on the west. It was an attempt to deal
with usurpation. Finding himself in a vulnerable position, Galerius
did not engage Maxentius, withdrew from Italy and proclaimed
Licinius as senior emperor in the west and Constantine as junior
emperor in the west. Maxentius was eventually defeated by
Constantine, who, after winning a civil war against Licinius (who
had become senior emperor of the eastern part of the empire after
the defeat of Maxentius) became the sole emperor of the whole Roman
Empire.
Theodosius I attacked Magnus Maximus a usurper emperor who took
over the provinces of the western part of the Roman Empire except
for Italy. Theodosius won. After this the emperor of the western
part of the empire, Valentinian II, was found hanging in his room
and there was another usurper emperor, Eugenius. Theodosius
defeated him and became the sole emperor of the whole of the Roman
Empire.
