12 problems

A bipartite circle graph is a circle graph whose vertices can be divided into two parts such that edges join only vertices in different parts. A graph $H$ is a pivot-minor of $G$ i…

Let $R_{\mathrm{piv}}(k)$ be the minimum $n$ such that every $n$-vertex graph contains an independent set or clique of size $k$ as a pivot-minor. Pivot-minor Ramsey conjecture. … T…

For a positive integer $s$, a graph is $K_{s,s}$-free if it contains no subgraph isomorphic to $K_{s,s}$. Linear pivot-minor degree-boundedness conjecture. For each bipartite graph…

Kwon–McCarty–Oum–Wollan's conjecture. Every pivot-minor-closed class of graphs has bounded rank-depth if and only if it does not contain $P_t$ or $K_t\boxslash K_t$ for some $t$.

A pivot-minor $H$ of a graph $G$ is proper if $|V(H)|<|V(G)|$. A graph $G$ is an intertwine of graphs $H_{1}$ and $H_{2}$ for pivot-minors if it contains both $H_{1}$ and $H_{2}$ a…

Path pivot-minor conjecture. For every path $P$, the class of $P$-pivot-minor-free graphs has bounded linear rank-width.

Caterpillar pivot-minor conjecture. For every caterpillar $T$, the class of $T$-pivot-minor-free graphs has bounded linear rank-width.

A graph $H$ is a bipartite circle graph if it is both bipartite and a circle graph. Oum's rank-width conjecture. For every bipartite circle graph $H$, there is an integer $r(H)$ su…

For a graph class $\mathcal G$, polynomially chi-bounded means that there is a polynomial $f$ such that every induced subgraph $J$ of every graph in $\mathcal G$ satisfies…

Let $H$ be a graph. A set of vertices $A$ is complete to a disjoint set $B$ if every vertex of $A$ is adjacent to every vertex of $B$, and anti-complete if no such edges exist. Str…

Let $H$ be a graph. For a graph $G$, let $\boldsymbol{\alpha}(G)$ be its maximum independent-set size and $\boldsymbol{\omega}(G)$ its maximum clique size. Pivot-minor Erdős–Hajnal…

For a graph $H$, a graph class has no pivot-minor isomorphic to $H$ if none of its members has such a pivot-minor. Pivot-minor conjecture. For every graph $H$, the class of graphs…