Russia said on Monday it was ready for dialogue with Petro Poroshenko, who is on course to become Ukraine's president, but warned the Kiev authorities not to step up armed operations against separatists in the east.
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Echoing remarks by President Vladimir Putin in the past few days, Foreign Minister Sergei Lavrov said genuine dialogue between Kiev and the east - where pro-Russian separatists have rebelled against the national leadership - was the key to resolving Ukraine's crisis.
Some of the separatist groups have responded to Poroshenko's apparent election by declaring martial law, escalating tensions.
Denis Pushilin, the leader of the self-proclaimed Donetsk People's Republic, declared martial law in interviews with Russian media and on his official Twitter account.
"The main task is to cleanse the republic's territory of the Ukrainian military," he told Russia's Itar-Tass state news agency.
The airport in the city of Donetsk was closed Monday after separatist gunmen showed up on its premises, local media reported.
Poroshenko won 53.7 per cent of the votes in Sunday's poll, the Central Election Commission said, with 55.75 per cent of the ballots counted. The pro-Western businessman clearly defeated former prime minister Yulia Tymoshenko, who came a distant second with about 13 per cent of the votes.
The turnout was roughly 60 per cent, according to election officials, but most polling stations in the insurgency-hit eastern regions of Donetsk and Luhansk remained closed.
Poroshenko said Sunday night that he would continue Ukraine's pro-Western foreign policy while the military operation against the pro-Russian separatists would continue. However, he also promised to make his first trip as president to the restive east, and he said he would seek a meeting with Russian President Vladimir Putin.
Pushilin claimed that dialogue with Poroshenko would hardly be helpful because the people in Donetsk were against all 21 candidates in Sunday's elections.
"Not one of them has condemned the Kiev authorities' criminal orders as a result of which civilians keep dying," he said.
The military campaign has killed about 100 people since it began in April.
In the capital, Kiev, former heavyweight boxing champion Vitali Klitschko won the mayor's race with 57.4 per cent of the vote, according to initial results.
Klitschko said late Sunday that he would like to see the barricades erected around the capital's Maidan, or Independence, Square during demonstrations against now-ousted president Viktor Yanukovych to be taken down.
"I'm grateful to the activists of Maidan, but now I call for them to clear the barricades," he told broadcaster 1+1.
The barricades have stayed up as protesters said they plan to keep a watchful eye on any Ukrainian government.
During three months of protests centred on Maidan, protesters declared the desire of many Ukrainians for closer ties to the West. The demonstrations were a key factor in destabilizing Yanukovych, who fled Ukraine in February.
Klitschko had already run twice for mayor of the Ukrainian capital - both unsuccessfully. The office has been rendered largely ceremonial since 2012 municipal reform, according to which the city is administered by a city manager who is appointed by the Ukrainian president.