On Visibility and Blockers

This expository paper discusses some conjectures related to visibility and blockers for sets of points in the plane

On the connectivity of visibility graphs

The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and vertex-connectivity of visibility graphs. Unless all its vertices are collinear, a visibility graph has diameter at most 2, and so it follows by a result of Plesn\'ik (1975) that its edge-connectivity equals its minimum degree. We strengthen the result of Plesn\'ik by showing that for any two vertices v and w in a graph of diameter 2, if deg(v) <= deg(w) then there exist deg(v) edge-disjoint vw-paths of length at most 4. Furthermore, we find that in visibility graphs every minimum edge cut is the set of edges incident to a vertex of minimum degree. For vertex-connectivity, we prove that every visibility graph with n vertices and at most l collinear vertices has connectivity at least (n-1)/(l-1), which is tight. We also prove the qualitatively stronger result that the vertex-connectivity is at least half the minimum degree. Finally, in the case that l=4 we improve this bound to two thirds of the minimum degree.Comment: 16 pages, 8 figure

Blocking Coloured Point Sets

This paper studies problems related to visibility among points in the plane. A point $x$ \emph{blocks} two points $v$ and $w$ if $x$ is in the interior of the line segment $\bar{vw}$. A set of points $P$ is \emph{$k$-blocked} if each point in $P$ is assigned one of $k$ colours, such that distinct points $v,w\in P$ are assigned the same colour if and only if some other point in $P$ blocks $v$ and $w$. The focus of this paper is the conjecture that each $k$-blocked set has bounded size (as a function of $k$). Results in the literature imply that every 2-blocked set has at most 3 points, and every 3-blocked set has at most 6 points. We prove that every 4-blocked set has at most 12 points, and that this bound is tight. In fact, we characterise all sets $\{n_1,n_2,n_3,n_4\}$ such that some 4-blocked set has exactly $n_i$ points in the $i$-th colour class. Amongst other results, for infinitely many values of $k$, we construct $k$-blocked sets with $k^{1.79...}$ points

Erd\H{o}s-Szekeres theorem for kk-flats

We extend the famous Erd\H{o}s-Szekeres theorem to $k$-flats in ${\mathbb{R}^d}

Graphs with four boundary vertices

18 pagesInternational audienceA vertex v of a graph G is a boundary vertex if there exists a vertex u such that the distance in G from u to v is at least the distance from u to any neighbour of v. We give a full description of all graphs that have exactly four boundary vertices, which answers a question of Hasegawa and Saito. To this end, we introduce the concept of frame of a graph. It allows us to construct, for every positive integer b and every possible ``distance-vector'' between b points, a graph G with exactly b boundary vertices such that every graph with b boundary vertices and the same distance-vector between them is an induced subgraph of G