Ukraine’s President Volodymyr Zelenskyy signed a presidential decree on Tuesday formally declaring the “impossibility” of holding negotiations with Russian President Vladimir Putin.

The decree backs a decision put forward by Zelenskyy's national security council and includes the point: “To declare the impossibility of conducting negotiations with the president of the Russian Federation, V. Putin.”

The decree echoed a statement made by Zelenskyy when Putin annexed Ukrainian territory last Friday, saying it showed it is impossible to negotiate with the current president.

-ABC News’ Patrick Reevell

Ukrainian forces on Sunday evening broke through part of Russia's defense of the disputed Kherson region, advancing from the region’s northeast into a territory Russia had claimed to annex as its own on Friday.

Ukrainian troops succeeded in pushing south along the Dnipro river, according to Ukrainian and Russian officials.

Russia's Defense Ministry on Monday partly confirmed the advance, saying Ukrainian forces "managed to drive a wedge deep into our defense."

It said Russian troops had fallen back to “pre-prepared lines of defense" and were using heavy artillery to halt a further Ukrainian advance. It claimed, without evidence, that Ukraine had suffered heavy losses, but acknowledged that Ukraine had an advantage in tank numbers there.

Russian military bloggers said on Sunday that Ukrainian troops advanced southwards in the direction of the village of Dudchany, several miles behind the rest of Russia's frontline in the region.

The advance raised questions about whether Russia would be able to hold the city of Kherson, the only regional capital it managed to seize in the invasion. For weeks, military experts have said Russia's position in the Kherson region has been deteriorating because Ukraine has destroyed the only bridges allowing Russia to re-supply its troops.

Kirill Stremousov, a Russian-installed official in the region, on social media acknowledged Ukrainian troops had advanced along the Dnipro towards Dudchany but claimed they had been halted by Russian fire and that “everything is under control.”

A continued Ukrainian advance along the Dnipro would threaten to undermine the rest of the Russian front north of the river, raising the risk Russian forces there could be cut off.

The White House National Security Council’s spokesman John Kirby noted Ukraine was making gains in the south on Monday, but caveated that they were “incremental” for the time-being.

The battle for Kherson has major military and symbolic significance for both sides. A retreat from the city would seriously undermine Russia's annexation of one of the four Ukrainian regions declared by Vladimir Putin just days ago -- Kherson is supposed to be the capital of the newly annexed region of the same name.

-ABC News' Patrick Reevell

The head of the Ukrainian nuclear power plant Zaporizhzhia has been released, after Ukrainian officials accused Russia of kidnapping him, according to the International Atomic Energy Agency.

Ihor Murashov, the head of the Zaporizhzhia nuclear power plant, was released and returned safely to his family, Rafael Mariano Grossi, the Director General of the IAEA, tweeted.

Zaporizhzhia is a Ukrainian facility now occupied by Russian troops.

Ukraine's nuclear power provider accused Russia on Saturday of "kidnapping" Murashov, according to the Associated Press.

Ukrainian state nuclear company Energoatom said Russian forces seized Murashov around 4 p.m. Friday. Energoatom said Russian troops stopped Murashov's car, blindfolded him and then took him to an undisclosed location, according to the AP.

Russian President Vladimir Putin's threats that his country could strike Ukraine with nuclear weapons were "irresponsible rhetoric" from a nuclear power, a Pentagon official said.

"They are continuing to be irresponsible rhetoric coming from a nuclear power," Pentagon spokesperson John Kirby said on "Good Morning America" on Monday. "There's no reason for him to use that kind of bluster, those kinds of threats."

But the U.S. was still taking the threats seriously, he said. The U.S. was "ready and prepared" to defend every inch of NATO territory, he said.

"We have to take these threats seriously. We must. It'd be easier if we could just blow it off, but we can't," Kirby said. "These are serious threats made by a serious nuclear power."