13 problems

Let $d53d$ be a proper vertex-minor-closed class of graphs. For $k,p displaymath in mathbb{N}$, a graph is $k$-rank-connected if it has at least $2k$ vertices and satisfies … f…

Let $G$ be an $n$-vertex circle graph, and let $\operatorname{CZ}(G)$ denote its CZ-distance. Circle-graph lower-bound conjecture. There exist $n$-vertex circle graphs $G$ with … T…

Let $G$ be a bipartite circle graph, namely a circle graph that is bipartite, and let $I(G)$ be its independence complex, whose simplices are the subsets of pairwise non-adjacent v…

Let $G$ be a circle graph, namely the intersection graph of the chords in a chord diagram, and let $I(G)$ be its independence complex, whose simplices are the subsets of pairwise n…

Przytycki–Silvero conjecture for bipartite circle graphs. If $G$ is a bipartite circle graph, then $I(G)$ has the homotopy type of a wedge of spheres.

Przytycki–Silvero conjecture. If $G$ is a circle graph, then $I(G)$ has the homotopy type of a wedge of spheres.

A circle graph is a graph that can be represented by chords of a circle, with vertices corresponding to chords and adjacency corresponding to chord intersection. The paired-dominat…

Paired-domination APX-hardness conjecture. The paired-domination problem is APX-hard on circle graphs.

Let $G$ be a circle graph, and let $I(G)$ be its independence complex. The size of the input is measured by the number of vertices of $G$. Polynomial-time homotopy-type conjecture.…

A circle graph is the intersection graph of chords of a circle. Let $r$ and $n$ be positive integers, and let the circle graph have $n$ vertices. The linear induced-subgraph conjec…

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…

Geelen's conjecture. These other Naji solutions might correspond in some way to splits.

A geometric graph is a graph drawn in the plane with vertices represented by points and edges as straight-line segments. A circle graph $F$ is the intersection graph of chords on a…