A police officer was killed and two others injured in the attack near the separatist stronghold of Slavyansk by pro-Russian rebels who launched a mortar attack on a checkpoint after taking up position in a church, deputy interior minister Serhiy Yarovy said.
Also near Slavyansk today, separatists shot down an An-26 military transport plane, a spokesman of the Ukrainian army was quoted as telling local media.
The aircraft was carrying humanitarian aid to the local population, Vladislav Seleznyov said according to Interfax Ukraine agency.
As violence continued throughout the nation's vast and heavily industrialised east, the authorities seemed to be losing control of the frontier with Russia.
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The border post at Severnyi, 70 kilometres from the city of Lugansk, was completely deserted today.
Pro-Russian fighters of the self-proclaimed "People's Republic of Lugansk" had ordered the Ukrainian border guards to leave, a local resident said.
Two border barriers had been lifted and anyone was able to cross.
The "People's Republic of Lugansk" claims to be in control of up to 200 kilometres of border. Ukrainian authorities admitted for the first time yesterday that they had been forced to abandon some checkpoints.
The porous border is likely to complicate efforts by Ukraine's government to re-impose its authority in the east.
The government in Kiev says truckloads of armed men and military equipment are attempting to cross every day to support the rebels.