Dallard–Krnc–Kwon–Milanič–Munaro–Štorgel–Wiederrecht forbidden-subgraph characterization

Let $\mathcal{F}$ be a finite class of graphs, and let $\mathcal{F}$-free mean having no induced subgraph isomorphic to a member of $\mathcal{F}$. A multiclaw is a graph each component of which is isomorphic to an induced subgraph of $K_{1,3}$; a subdivided multiclaw is a subdivision of such a graph, and its line graph has one vertex for each edge with adjacency inherited from edge incidence. Dallard–Krnc–Kwon–Milanič–Munaro–Štorgel–Wiederrecht's conjecture. The class of all $\mathcal{F}$-free graphs has bounded $\operatorname{{\text{\sf{tree}}-\alpha}}$ if and only if there exist $F_1,F_2,F_3\in\mathcal{F}$ where $F_1$ is complete bipartite, $F_2$ is a subdivided multiclaw, and $F_3$ is the line graph of a subdivided multiclaw. The paper identifies this as an equivalent formulation of the finite-forbidden-subgraph conjecture and then proves the conjecture.