### Ideal analogues of some variants of Hurewicz property

#### Abstract

In this paper, we continue the study on the ideal analogues of several variations of Hurewicz property introduced by Das et al. \cite{Das,DKC,DCS} for example, $ \mathcal{I} $-Hurewicz ($ \mathcal{I} $H), star $ \mathcal{I} $-Hurewicz (S$ \mathcal{I} $H), weakly $ \mathcal{I} $-Hurewicz (W$ \mathcal{I} $H) and weakly star-$ \mathcal{I} $-Hurewicz (WS$ \mathcal{I} $H). It is shown that several implications in the relationship diagram of their concepts are reversible under certain conditions, for instance; (1) If paracompact Hausdorff space has the WS$ \mathcal{I} $H property then it has W$ \mathcal{I} $H property. (2)If the complement of dense set has $ \mathcal{I} $H property then W$ \mathcal{I} $H property implies $ \mathcal{I} $H property and(3)If the complement of dense set has S$ \mathcal{I} $H property then WS$ \mathcal{I} $H property implies S$ \mathcal{I} $H property. Examples of spaces are given which have W$ \mathcal{I} $H property but not separable. In addition we introduced the ideal analogues of some new variations of Hurewicz property called mildly $ \mathcal{I} $-Hurewicz and star K-$ \mathcal{I} $-Hurewicz properties and explore their relationship with other variations of $ \mathcal{I} $-Hurewicz property. We also study the preservation under certain mappings.