The Yemeni government launched a fresh offensive in August 2009
Yemeni government forces fighting Houthi rebels in the country's north have killed a leader and forced his supporters into retreat, officials say.
A government website quoted security sources as saying that Ali al-Qatwani was killed when troops took control of the al-Mahaleet area of Saada province.
Two other Houthi commanders were killed in fighting on Wednesday, it reported.
A Saudi soldier was shot dead the day before in clashes with the rebels on the border with Yemen, Saudi media say.
Issa Madkhali was killed in the mountainous Jabal al-Dukhan area, which straddles the frontier, al-Hayat reported.
Riyadh launched an offensive against the Houthis this month after they occupied villages inside Saudi territory and killed a border guard.
It has warned that air strikes and shelling inside Yemen will continue until the rebels withdraw tens of kilometres from the border.
Yemeni government forces have also intensified their assault on rebel strongholds, and commanders say they are getting close to regaining full control of several key strategic locations in Saada.
The Houthis say they have destroyed a number of Yemeni army tanks
Abbas Aid, head of a rebel combat unit, and Abu Haidar, another senior figure, were killed on Wednesday, security sources said. Youssef al-Madani, the son-in-law of Hussein al-Houthi, founder of the rebel group, was allegedly wounded.
On Thursday, the armed forces and local authorities in Saada called on rebel fighters to hand themselves in.
"Give yourself up... and we will not do you harm," a statement said.
On their website, the rebels said their fighters had destroyed three Yemeni army vehicles and disabled two tanks near Harf Sufian.
They also accused the Saudi military of bombing homes, government buildings and a market in al-Mahalit and Amran provinces.
The Houthis, named after the family of their leader, say they are trying to reverse the political, economic and religious marginalisation of the Zaydi Shia community.
They also accuse Saudi Arabia of supporting the Yemeni armed forces by allowing them to launch attacks from its territory, a charge both countries deny.
The Yemeni government accuses the Houthis of wanting to re-establish Zaydi clerical rule, which ended in 1962.
The Zaydi community are a minority in Yemen, but make up the majority in the north of the country.
The insurgents first took up arms against the government in 2004, after which government forces killed or captured much of the Houthi leadership.
The government launched a fresh offensive in August 2009, which has precipitated a new wave of intense fighting.
Aid agencies say tens of thousands of people have been displaced.