§ 44-b. Necessary parties; lien against public or private improvement. Notwithstanding any inconsistent provision of section forty-four of this article, any private owner or the state or a public corporation with which a notice of lien is filed shall not be a necessary party defendant in an action to enforce the lien if, either before or after the commencement of the action, a contractor or subcontractor, (a) in the case of a public improvement, executes a bond or undertaking, in accordance with subdivision five of section twenty-one of this chapter, to the state or the public corporation with which the notice of lien is filed conditioned for the payment of any judgment that may be recovered in an action to enforce the lien or, (b) in the case of a private improvement, executes a bond or undertaking in accordance with subdivision four of section nineteen of this chapter, to the county clerk with which the notice of lien is filed conditioned for the payment of any judgment that may be recovered in an action to enforce the lien.