36 problems

Big-line Big-clique Conjecture. For any $k\geq 3$ and $\ell\geq 3$, there is an integer $n=n(k,\ell)$ such that every finite set of at least $n$ points in the plane contains either…

A point visibility graph is the graph associated with a finite set of points in the plane, with two points adjacent when they are visible to each other in the relevant geometric se…

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…

Non-SCRVG conjecture. The construction of a maximal closed RIG given in the cited work is not an SCRVG.

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…

Juntarapomdach–Kittipassorn conjecture. The graph $D_n^2$ is a non-TRVG for every $n\geq 10$.

Let $K$ be a closed Jordan curve, let $P\subset K$ be a set of points, and let the visibility graph of $P$ with respect to $K$ be the graph whose vertices are the points of $P$, wi…

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.

Let $G$ be a rectangle visibility graph, and let $\operatorname{height}(G)$ denote the minimum height among its rectangle-visibility representations. Height-two rectangle visibilit…

Hliněný et al.'s conjecture. FO model checking is FPT on weak visibility graphs of simple polygons, additionally parameterized by the independence number.

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 $\hat G$ be an undirected graph, let $G$ be an orientation of $\hat G$, and let $b(G)$ and $b(\hat G)$ denote their bar visibility numbers. The orientation bound conjecture. ……

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 $G$ be a graph with at most $10$ vertices. Let $\operatorname{obs}_{\mathrm{in}}(G)$ be the number of obstacles needed to represent $G$ in the absence of outside obstacles, and…

Let $G$ be a graph whose circumference is at most $7$, where the circumference is the length of a longest cycle in $G$. An outside-obstacle representation represents $G$ as a visib…

Let $G$ be a graph, let $\operatorname{obs}(G)$ be the smallest number of obstacles needed to represent $G$ as a visibility graph, and let $\operatorname{obs}_{\mathrm{out}}(G)$ be…

Let $G$ be a graph. Write $\operatorname{obs}(G)$ for the smallest number of obstacles needed to represent $G$ as a visibility graph, and let $\operatorname{obs}_{\mathrm{out}}(G)$…

Babbitt et al.'s conjecture. The complete graph $K_{3k+4}$ is not a semi-arc $k$-visibility graph.

Let $X$ be a finite set of points in the Euclidean plane with at most $\ell$ collinear points, and let $G_X$ be its visibility graph. Define $mc_\ell(k)$ to be the maximum cardinal…