Excluding a Ladder as an Induced Minor in Graphs Without Induced Stars

A $k$-ladder is the graph obtained from two disjoint paths, each with $k$ vertices, by joining the $i$th vertices of both paths with an edge for each $i\in\{ 1,\ldots,k\}$. In this paper, we show that for all positive integers $k$ and $d$, the class of all $K_{1,d}$-free graphs excluding the $k$-ladder as an induced minor has a bounded tree-independence number. We further show that our method implies a number of known results: We improve the bound on the tree-independence number for the class of $K_{1,d}$-free graphs not containing a wheel as an induced minor given by Choi, Hilaire, Milani\v{c}, and Wiederrecht. Furthermore, we show that the class of $K_{1,d}$-free graphs not containing a theta or a prism, whose paths have length at least $k$, as an induced subgraph has bounded tree-independence number. This improves a result by Chudnovsky, Hajebi, and Trotignon. Finally, we extend the induced Erd\H{o}s-P\'osa result of Ahn, Gollin, Huynh, and Kwon in $K_{1,d}$-free graphs from long induced cycles to any graph that is an induced minor of the $k$-ladder where every edge is subdivided exactly once.