The Scottish First Division side reached this stage after their run to the Scottish Cup final.
Robertson was sacked on Sunday after a disappointing first season back in the Scottish First Division.
Tuffey was a regular in the Partick side that just missed out on promotion from the Scottish First Division.
Mowbray signed Graham Dorrans from the Scottish First Division club last summer and the midfielder has recently featured in the Baggies first-team.
Bowman made the first of his two first-team appearances for Falkirk in March but has not played for the Scottish First Division side this season.
Dorrans, who had been with Livi since the age of 13 and still had 18 months of his contract still to run, is the Scottish First Division club's top scorer this season with 10 goals.
Caley Thistle enjoyed a run of 21 matches unbeaten as they won promotion from the Scottish First Division but found in their re-introduction to the Scottish Premier League, against Celtic, that things were going to be tougher.
There was also a Scottish Cup exit at the hands of First Division Clyde.
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Despite a numbing Scottish Cup semi-final defeat by First Division outfit Ross County, only his third match in charge, Lennon did enough to convince the Celtic board he was the man to succeed Mowbray.
The Dons were looking to avenge their humiliating Scottish Cup fifth-round replay defeat by the First Division side in February, but had their work cut out to avoid another cup shock.
The only black mark against him was an embarrassing Scottish Cup semi-final defeat at the hands of First Division Ross County.
The Scottish beer division's operating profit increased by 20% in the first half of the financial year, on a 7% rise in revenues.
The Pittodrie club cannot be relegated to the First Division although manager Ebbe Skovdahl wants his side to finish strongly and boost their Scottish Cup Final dreams.
And a number of First Division clubs are pressing for a breakaway league to become a second-tier to the Scottish Premier League, which set up its own separate structure some years ago.
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