What would it take to demolish Boris Johnson’s majority in parliament? After the Liberal Democrats won the Brecon and Radnorshire byelection, the new prime minister’s working majority is now just one. But is the parliamentary arithmetic deceptive – and what would it take to make it zero?

Conservative rebels

We already know there are enough Conservative rebels to sink any legislation that would propel the UK towards no deal or to pass measures to prevent it from happening or to obstruct the prorogation of parliament.

Before recess, 17 Conservatives voted for Dominic Grieve’s amendment to the Northern Ireland bill, which was aimed at preventing prorogation, including the former ministers Alistair Burt, Sarah Newton, Margot James and Steve Brine.

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Newly backbench former cabinet ministers and ministers include the no-deal sceptics Philip Hammond, David Gauke, Greg Clark, David Mundell, Stephen Hammond and Tobias Ellwood.

It remains highly unpredictable what these MPs would do in a confidence vote if that was the only way of preventing no deal. Grieve and Ken Clarke are the only MPs who have said they would be willing to vote down their own government – and that may not be enough if rebels in other parties decide to go another way.

There is also the question of pro-Brexit rebels – the ones who sank Theresa May’s premiership – though many of them such as Priti Patel, Dominic Raab, Esther McVey and Jacob Rees-Mogg are now in Johnson’s cabinet.

There are some hard Eurosceptics such as Steve Baker still out in the cold, however, and one of them, Mark Francois, has suggested there are 60 pro-Brexit rebels willing to torpedo the withdrawal agreement even if the backstop is removed.

That is likely to be an overestimate – only 34 voted against May’s deal at the final vote – but it could be the case that Brexiters scupper any “great deal” that Johnson claims he can bring back.

Labour rebels

There are between 10 and 20 Labour MPs who are determined to see through Brexit. Some are pro-leave themselves and others such as Caroline Flint, Melanie Onn, Gareth Snell and Sarah Champion want to stop a second referendum because of the strength of feeling among their pro-leave constituents.

Astonishingly, very few of these MPs voted for May’s Brexit deal, though it is highly likely more would vote for a deal if one was presented to parliament in the coming months. It is unclear if they would be enough in number for the deal to pass.

Most Labour rebels, even if they want to deliver Brexit, are squeamish about a no deal. Some hardliners, such as Kate Hoey, would be prepared to oppose moves to block no deal, and views among some others are hardening. Champion has said she would not block no deal.

However, it is likely none of these Labour MPs who might rebel on Brexit would be prepared to back the government in a confidence vote. To vote confidence in a Johnson government would be to end their careers and have the Labour whip removed. Hoey, who has said she will stand down at the next election, is the only real possibility.

Independents and Change UK

The House of Commons has acquired a rag-tag bunch of 17 independent MPs over the past two years and with such a tiny majority, their votes could make all the difference.

One group of independents are passionately pro-remain, some of them former Tory and Labour MPs who left and formed Change UK and then left that fledgling party. There are five in that category: Heidi Allen, Sarah Wollaston, Luciana Berger, Angela Smith and Gavin Shuker. All of them would vote to block no deal, and although a confidence vote would probably lead to them losing their seats, they would probably be willing to vote down the government.

The remaining five members of what is now the Independent Group for Change, including Chris Leslie and Anna Soubry, who have previously said they would vote with the government in a confidence vote, may also vote down Johnson if the alternative is no deal.

Charlie Elphicke, the former Tory MP who is currently suspended from the party while charged with sexual assault, is likely to vote with the government. Indeed, several commentators have claimed it is wrong to consider Johnson’s majority as one since Elphicke is likely to act as a Tory. Likewise, the suspended Labour MP Chris Williamson is likely to follow the Labour whip.

Three former Labour MPs who quit their parties in anger at the leadership – Frank Field, Ian Austin and John Woodcock – feel so strongly opposed to Jeremy Corbyn that they could vote with Johnson in a confidence vote, and Woodcock did vote with the government last time around.

Others are less predictable, including the former Labour MP Jared O’Mara, who has been largely absent from parliament and says he intends to quit in September, and Sylvia Hermon, the Northern Irish independent MP, who is strongly opposed to no deal but has previously voted with the government in a confidence vote.

The DUP

Johnson has spent many hours since he entered Downing Street trying to woo back the Democratic Unionist party, and its leaders have expressed confidence in his pursuit of an EU exit, deal or no deal, by 31 October.

Johnson’s new Northern Ireland secretary, Julian Smith, is likely to be able to keep them onboard for now and they should be dependable votes for the government if a confidence vote is triggered. It is less predictable how they would move if there is some new deal presented by Johnson that contains any elements of the backstop.

Future byelections

Though there could be byelections looming in O’Mara’s seat of Sheffield Hallam and Elphicke’s seat of Dover, only one of them has any chance of causing Johnson to lose his majority. Hallam has historically been a three-way marginal but Nick Clegg’s former seat is likely to be a straight fight between Labour and the Lib Dems next time.

In Dover, however, the Brexit party may fancy the opportunity to win a seat off the Conservatives, though the reality is that any byelection in that part of Kent is not a certainty and would take place well past 31 October, given Elphicke’s trial has not yet commenced.