The Ukrainian government has captured most of a town near Donetsk, tightening the noose around the key rebel-held stronghold, officials said on Tuesday.

The Ukrainian government has captured most of a town near Donetsk, tightening the noose around the key rebel-held stronghold, officials said on Tuesday.

Interior Minister Arsen Avakov said in a Facebook post that one soldier was killed and four wounded when a volunteer battalion came under mortar fire before entering the town of Ilovaysk, 18 kilometers (11 miles) east of Donetsk. Among the wounded is the commander of the Donbass battalion, Semyon Semenchenko, who said government soldiers destroyed three rebel checkpoints and four firing positions, and that fighting continues.

Government efforts to quell the pro-Russian separatist front have focused lately on gradually encircling Donetsk, the largest rebel-controlled city.

Volunteer battalions have proven essential in the government offensive against armed separatists in the mainly Russian-speaking east.

Fighting between government troops and pro-Russian separatists in eastern Ukraine has forced nearly 344,000 people to flee their homes, according to the United Nations — a number that has only grown as the humanitarian situation in rebel-held cities deteriorates.

The situation appears to be particularly dire in the city of Luhansk, near the Russian border, which has been left without electricity, running water or phone connections for 17 days. Independent reports from the besieged city are scarce since the area is very dangerous for journalists.

The city hall said in a statement on its website on Tuesday that Luhansk and its city center came under fierce shelling overnight. An unspecified number of civilians were killed or wounded. Residents are standing in lines to buy bread as food supplies are running out, the city hall said. Authorities also raised the alarm about a potential outbreak of infectious diseases since household waste has not been taken out of town for more than two weeks amid sweltering heat.

Ukraine and the West have voiced concern about Russia’s military activity at the Ukrainian border. Moscow has said it can do what it wants within its borders but invited a mission from the Organization for Security and Cooperation in Europe to observe two border crossings in a bid to counter claims that the Kremlin is supplying the rebels with weapons.

Paul Picard, head of that mission, told reporters in a Russian border town on Tuesday that observers had seen a marked increase of military activity around the border points over the past week, including Russian helicopter activity. But he said no helicopters were observed crossing the border.

Picard also said that the observers had seen “groups of young men and women wearing military-style dress” moving back and forth across the border. No weapons or military vehicles were observed.

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Nataliya Vasilyeva in Kiev, Ukraine, and Jim Heintz in Moscow contributed to this report.