Afghan national forces took over full responsibility for the country’s security Jan. 1 as NATO-led coalition forces ended their combat mission after 13 years. (Jawad Jalali/European Pressphoto Agency)

Many Afghans would like to see a greater American role after this year than is planned, while a majority believe that last year’s runoff election that led to a U.S.-brokered power-sharing government was “mostly fraudulent,” according to a national poll released Thursday.

The poll, conducted in November by the Afghan Center for Socio-Economic and Opinion Research in collaboration with New York-based firm Langer Research Associates, surveyed a random sample of 2,051 Afghan men and women in most parts of the country.

In the poll, 46 percent said they want to see a greater commitment by U.S. forces than is in place. The United States and its NATO allies withdrew most of their forces last year and have roughly 13,000 troops in Afghanistan. The U.S. contingent is expected to shrink to about 5,000 by the end of the year.

In recent weeks, senior Afghan officials have also indicated that they would like to see a greater U.S. presence after this year. Only 29 percent of Afghans said they prefer that fewer or no U.S. troops remain, according to the poll, which has a margin of error of plus or minus three percentage points. Two-thirds of Afghans favor a significant role for U.S. and international forces in training Afghan forces in the future, the poll found.

Six in 10 rated their security against crime and violence as at least “somewhat good” in November. But concerns about growing violence and insecurity may have increased in recent weeks as the Taliban Islamist insurgency mounted attacks in the capital, Kabul, and in other areas.

The latest assaults came Thursday, when a roadside bomb killed a police commander in Laghman province in eastern Afghanistan. Hours later, a suicide bomber detonated his explosives belt at the man’s funeral, killing four policemen and 12 civilians and wounding scores, Afghan officials said.

“This is against all Islamic principles,” said Gen. Naqibullha, a lawmaker from Laghman, who like many Afghans uses only one name. “The enemies of Afghanistan don’t even let us bury our dead peacefully.”

Although 53 percent of Afghans think the runoff election was mired in fraud, the poll found that an overwhelming majority approve of the unity government led by President Ashraf Ghani and his chief executive, Abdullah Abdullah. Three-quarters of Afghans see each as holding office legitimately. In the survey, 61 percent said Afghanistan is heading in the right direction, up from 48 percent in 2013.

Most Afghans, as expected, found persistent problems with basic services such as electricity, and they cited high unemployment and widespread corruption among government officials and police as major concerns, the poll found.

There were also troubling trends. The number of Afghans who view opium cultivation as “unacceptable in all circumstances” fell to 53 percent, the lowest on record, according to the poll. With widespread poverty, opium production reached record high levels last year.

Although the Taliban remains extremely unpopular, tolerance for it has grown. A fifth of Afghans surveyed said they support the presence of Taliban fighters, twice the number who said so in 2010, the poll found. And 31 percent said they support the presence of foreign jihadists, also an increase from 2010.

It is unclear whether such sentiments indicate a radicalization in some areas or whether the respondents felt more secure in Taliban-controlled areas. A sizeable majority support negotiating with the Taliban to find a political settlement if the Taliban agrees to stop fighting.

Large numbers said they would accept the Taliban’s restrictions on women’s rights in exchange for such a deal. Two-thirds would accept a ban on women leaving their homes unescorted, working, holding political office, attending school and voting.

Mohammad Sharif contributed to this report.