Frank formally meets Dunbarton at the football game he takes Bill to. While there, Dunbarton seems to take a shine towards Frank, who earned his respect. He says he's sympathetic towards the union's plot, but is actually unfazed by it.
Dunbarton is a generally greedy, unsympathetic and duplicitous individual. For example, when Ed was killed he was more concerned with the cost of replacing the propeller that caused his death. He is rude and dismissive to those he considers beneath him, especially if he thinks they are not respecting his wife. Ironically though, he is rude to his wife too, such as when he yells at her not to buy a hot dog because she's getting fat.
Eventually after solving the strike, Dunbarton was unimpressed with Frank's compassion and fired him on Christmas Eve. After firing Frank, Dunbarton brought in his nephew, former Minor League Baseball player Scoop Dunbarton, as Frank's replacement.
In the Season 2 finale "Landing The Plane", Dunbarton discovers that Frank was responsible for trying to rid the airport of his nephew. Dubarton then threatens to have him "rot in a dungeon" while his kids work at a Cambodian sweatshop to make him cheap clothing that he'll only wear once.
Scoop steps on a suitcase bomb set up by terrorists who attempted to hijack a plane, which led to Scoop being blown up and the laminated newspaper article of him (which he always wore around his neck) to fly across the air, and stab Dunbarton in the neck, killing him almost.