Livingston manager David Martindale said he faces a "conundrum" over his future after the Scottish Premiership bottom club's "horrendous season" continued with a home thrashing by Dundee.
The West Lothian side are now winless in 14 league games and remain six points adrift at the foot of the table.
Amadou Bakayoko's strike gave Dundee the lead that was extended by Luke McCowan after home skipper Andrew Shinnie had a penalty saved.
There were moments of hope for the hosts as McCowan was sent off and Tete Yengi scored a spot-kick to reduce the deficit, but late goals from Zach Robinson and impressive debutant Michael Mellon made sure of the victory that lifts Dundee to seventh.
"I am conscious that I am the leader of the group," said Martindale, whose side have not won in the league since 7 October.
"It's not good enough. We are away. We are in the Championship next season at this rate.
"It's not good enough from me. I think I have got to take responsibility. It lies with me. I genuinely believed we could get the club out of this situation.
"I don't want to walk out on the players or the staff. But I have to take responsibility. That's the conundrum I am in. I don't walk away from fights, we are in a fight."
Livingston, desperate to end a six-game winless run at home in the league, started strong and threatened Harrison Sharp's goal twice early on.
Against the run of play however, Bakayoko netted his fifth goal of the campaign, drilling beyond debutant Michael McGovern after James Penrice's hacked clearance fell fortunately to the Dundee number nine.
A chance arose for the home side to equalise when McCowan felled Jason Holt in the box, but Shinnie's weak spot-kick was kept out by Sharp.
A flurry of changes at the hour from both sides ripped the game wide open. A disorganised Livingston defence were passed through with ease by McCowan and Burnley loanee Mellon, allowing the former to coolly slot in Dundee's second.
An incredible five-minute spell caused panic for Dundee. McCowan was sent off for a second bookable offence and Yengi netted from the spot soon after.
But, immediately from kick-off, Robinson turned in a Mellon cross to restore the visitors' two-goal margin.
Mellon was not finished yet, however. The new arrival endeared himself further to the travelling support as he struck from 20 yards to settle the game.
Player of the match - Michael Mellon (Dundee)
Clinical Dundee rejoice as Livingston run out of ideas - analysis
Livingston's miserable winless league run continues despite last weekend's rare victory in the Scottish Cup.
Martindale discussed a change of shape pre-match and it appeared to have the desired effect - for almost 20 minutes at least before Bakayoko slammed home the opener.
Livingston failed to get forward enough to support Bruce Anderson in attack until desperate gung-ho attacking substitutions were made.
These changes momentarily forced a couple of errors from an overwhelmed Dundee defence. Amateurish defending of their own handed Dundee the victory on a plate, however, once the visitors settled following McCowan's red card.
Tony Docherty's side grew into the game after a slow start and never looked like dropping points - even after the dismissal and Yengi's spot-kick.
It would be easy to pin the performance on Livingston's failings, but credit must be given to a well-oiled team who showed class throughout and were ruthless in front of goal.
Sharp impressed in goal with numerous fine stops, while the backline and midfield five refused to yield any space to the home side - even with only 10 men on the park.
What they said
Livingston manager David Martindale: "If I walk, what message does that send to them? But as a group, with me as the figurehead, it's been shocking. A horrendous season."
Dundee manager Tony Docherty: "In the first half we never quite got to grips with the game - maybe it was the change of shape. In the second half there was real resilience. I asked my players to stand up today and they did. I'm really proud of them."
Livingston host Ross County and Dundee travel to Aberdeen, both on Tuesday at 19:45 GMT.