7 problems

Let $k\ge 3$, let $\varepsilon>0$, and let $G$ be an $n$-vertex incomparability graph. Here $\varrho_2(G)$ denotes the edge density of $G$, and $K_k[t]$ denotes the complete $k$-pa…

The co-bipartite forbidden-subgraph conjecture. For any co-bipartite $H$, there is an output-polynomial time algorithm for $\textsc{Dom-Enum}$ in $H$-free incomparability graphs.

The $2K_p$-free incomparability conjecture. For every $p$, there is an output-polynomial time algorithm for $\textsc{Dom-Enum}$ in $2K_p$-free incomparability graphs.

The $S_t$-free incomparability conjecture. For every $t$, there is an output-polynomial time algorithm for $\textsc{Dom-Enum}$ in incomparability graphs of $S_t$-free posets.

Clique-separator conjecture. There is a clique cover $C$ in $G$ such that removing

Shallow-minor star conjecture. If $G$ does not have an induced star on $s$ leaves, then, for every $t\ge 0$,

Let $\mathcal{P}=(P,\leq)$ be a poset of width $w$, and let $\chi_{\mathrm{g}}(a,b,w)$ denote the maximum of the $(a,b)$-game chromatic number over all posets of width $w$. Finiten…