21 problems

A semi-arc $k$-visibility graph is a graph represented by semi-arcs, with visibility allowing at most $k$ obstructing semi-arcs; let $n$ be the number of vertices. Semi-arc $k$-vis…

Big-Line-Big-Clique Conjecture. For all integers $t$ and $\ell$ there is an integer $n$ such that every finite set $P$ of at least $n$ points in the plane either contains $\ell$ co…

A graph is representable as a visibility graph if it admits a visibility representation by regions whose unblocked sightlines correspond exactly to the edges of the graph. Connecte…

Let $G$ be a planar graph. For each internal face $F$ in a plane drawing of $G$, let $G_F$ denote the subgraph induced by the vertices incident with $F$. Compact visibility conject…

Let $A\subset\mathbb{R}^2$ be a finite point set. A visibility patch is a subset $W\subset\Gamma(\mathbb{R})$ of a real cubic on which the relevant pairs of points are considered,…

Let $A\subset\mathbb{R}^2$ be a finite planar point set. Two points of $A$ are visible if the open segment between them contains no other point of $A$; a mutually visible subset is…

The $D^2_n$ non-TRVG conjecture. $D^2_n$ is a non-TRVG when $n\ge10$.

The $D^a_n$ non-TRVG conjecture. $D^a_n$ is a non-TRVG when $2a+5\le n\le 2a+7$ and $a\ge 3$.

Minimum-edge non-TRVG conjecture. $K_{3,5}$ is a non-TRVG with the least number of edges.

Neighborhood-island conjecture. The neighborhood of some point $p\in P$ contains an island of size $f(n)$, where $f(n)$ tends to infinity as $n$ tends to infinity.

Let $T_n$ denote the transitive tournament on $n$ vertices, and let $b(G)$ be the bar visibility number of a digraph $G$. The conjecture. … This was one of two conjectures posed by…

An ordered curve pseudo-visibility graph is the ordered graph arising from the pseudo-visibility representation by curves discussed in the paper. Recognition conjecture. Ordered cu…

An ordered graph is a graph equipped with a vertex order. An ordered graph $(G,\prec)$ is capped if, for every four vertices $a\prec b\prec c\prec d$, the edges $ac$ and $bd$ imply…

Let $X$ be a finite set of points in the Euclidean plane. For $u,v\in X$, let $(uv)$ denote the open line segment with endpoints $u$ and $v$, and let the visibility graph $G_X$ be…

Let $K_{3k+4}$ denote the complete graph on $3k+4$ vertices, and let a semi-arc $k$-visibility graph be a graph represented by semi-arcs with visibility determined by at most $k$ b…

Bar 1-visibility conjecture. Every 1-planar graph is a bar 1-visible graph.

Let $P$ be a $k$-blocked point set, with its points assigned one of $k$ colours so that two distinct points have the same colour exactly when some other point of $P$ blocks them. C…

A finite point set in the plane is $k$-blocked if its points are assigned one of $k$ colours such that two distinct points have the same colour exactly when some other point of the…

For fixed $\ell$, let $b_\ell(n)$ be the minimum integer such that every set of $n$ points in the plane with no $\ell$ collinear points can be blocked by a set of $b_\ell(n)$ point…

Let $P$ be a finite set of points in the plane, and let $\mathcal{V}(P)$ be its visibility graph. A $(k-1)$-colouring assigns one of $k-1$ colours to each point. Big-Line-Big-Chrom…

Big line or big clique conjecture. For all integers $k$ and $\ell$ there is an integer $n$ such that every finite set of at least $n$ points in the plane contains either $\ell$ col…