A subset $M$ of vertices in a graph $G$ is a mutual-visibility set if any two vertices $u$ and $v$ in $M$ ``see'' each other in $G$, that is, there exists a shortest $u,v$-path in $G$ that contains no elements of $M$ as internal vertices. The mutual-visibility number $μ(G)$ of a graph $G$ is the largest size of a mutual-visibility set in $G$. Let $n\in\mathbb{N}$ and $Q_{n}$ be an $n$-dimensional hypercube. Cicerone, Di Fonso, Di Stefano, Navarra, and Piselli showed that $2^{n}/\sqrt{n}\leqμ(Q_{n})\leq2^{n-1}$. In this paper, we prove that $μ(Q_{n})>0.186\cdot2^n$ and thus establish that $μ(Q_{n})=Θ(2^{n})$.
We also consider the chromatic mutual-visibility number, $χ_μ(G)$, defined as the smallest number of colors used on vertices of $G$, such that every color class is a mutual-visibility set in $G$. Klavžar, Kuziak, Valenzuela-Tripodoro, and Yero asked whether $χ_μ(Q_{n})=O(1)$. We answer their question in the negative, namely, we show that $χ_μ(Q_{n})$ is a growing function of $n$. Moreover, we show that $χ_μ(Q_{n})=O(\log\log{n})$.
Finally, we study the so-called total mutual-visibility number of graphs and give asymptotically tight bounds on this parameter for hypercubes.
A visibility representation of a graph $G$ is an assignment of the vertices of $G$ to geometric objects such that vertices are adjacent if and only if their corresponding objects are "visible" each other, that is, there is an uninterrupted channel, usually axis-aligned, between them. Depending on the objects and definition of visibility used, not all graphs are visibility graphs. In such situations, one may be able to obtain a visibility representation of a graph $G$ by allowing vertices to be assigned to more than one object. The {\it visibility number} of a graph $G$ is the minimum $t$ such …
Independent mutual-visibility sets and distance edge-critical graphs
Zhejiang University of Science and Technology（浙江科技学院）；University of Ljubljana（卢布尔雅那大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；University of Maribor（马里博尔大学）
If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P) \cap X \subseteq \{x,y\}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $μ_{\rm t}(G)$ of $G$. Graphs with $μ_{\rm t}(G) = 0$ are characterized as the graphs in which no vertex is the central vertex of a convex $P_3$. The total mutual-visibility number of Cartesian products is bounded and several exact results proved. For instance, $μ_{\rm t}(K_n\,\square\, K_m) = \max…
Given a graph $G=(V(G), E(G))$ and a set $P\subseteq V(G)$, the following concepts have been recently introduced: $(i)$ two elements of $P$ are \emph{mutually visible} if there is a shortest path between them without further elements of $P$; $(ii)$ $P$ is a \emph{mutual-visibility set} if its elements are pairwise mutually visible; $(iii)$ the \emph{mutual-visibility number} of $G$ is the size of any largest mutual-visibility set. % In this work we continue to investigate about these concepts. We first focus on mutual-visibility in Cartesian products. For this purpose, too, we introduce and in…
We study the set of visible lattice points in multidimensional hypercubes. The problems we investigate mix together geometric, probabilistic and number theoretic tones. For example, we prove that almost all self-visible triangles with vertices in the lattice of points with integer coordinates in $\mathcal W=[0,N]^d$ are almost equilateral having all sides almost equal to $\sqrt{d}N/\sqrt{6}$, and the sine of the typical angle between rays from the visual spectra from the origin of $\mathcal W$ is, in the limit, equal to $\sqrt{7}/4$, as $d$ and $N/d$ tend to infinity. We also show that there e…
A variety of the mutual-visibility coloring problem for graphs
Cochin University of Science and Technology（科钦科技大学）；University of Maribor（马里博尔大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；Universidad de Cádiz（加的斯大学）
Let $G=(V,E)$ be a graph and $P\subseteq V$ a set of points. Two points are mutually visible if there is a shortest path between them without further points. $P$ is a mutual-visibility set if its points are pairwise mutually visible. The mutual-visibility number of $G$ is the size of any largest mutual-visibility set. In this paper we start the study about this new invariant and the mutual-visibility sets in undirected graphs. We introduce the mutual-visibility problem which asks to find a mutual-visibility set with a size larger than a given number. We show that this problem is NP-complete, w…
It is established that the logarithm of the number of latin $d$-cubes of order $n$ is $Θ(n^{d}\ln n)$ and the logarithm of the number of pairs of orthogonal latin squares of order $n$ is $Θ(n^2\ln n)$. Similar estimations are obtained for systems of mutually strong orthogonal latin $d$-cubes. As a consequence, it is constructed a set of Steiner quadruple systems of order $n$ such that the logarithm of its cardinality is $Θ(n^3\ln n)$ as $n\rightarrow\infty$ and $n\ {\rm mod}\ 6= 2\ {\rm or}\ 4$.
We study in detail the so-called looping case of Mozes's game of numbers, which concerns the (finite) orbits in the reflection representation of affine Weyl groups situated on the boundary of the Tits cone. We give a simple proof that all configurations in the orbit are obtainable from each other by playing the numbers game, and give a strategy for going from one configuration to another. The strategy gives rise to a partition of the finite Weyl group into finitely many graded posets, one for each extending vertex of the associated extended Dynkin diagram. These are selfdual and mutually isomo…
A graph is called a strong (resp. weak) bar 1-visibility graph if its vertices can be represented as horizontal segments (bars) in the plane so that its edges are all (resp. a subset of) the pairs of vertices whose bars have a $ε$-thick vertical line connecting them that intersects at most one other bar. We explore the relation among weak (resp. strong) bar 1-visibility graphs and other nearly planar graph classes. In particular, we study their relation to 1-planar graphs, which have a drawing with at most one crossing per edge; quasi-planar graphs, which have a drawing with no three mutually …
This thesis focuses on two concepts which are widely studied in the field of computational geometry. Namely, visibility and unit disk graphs. In the field of visibility, we have studied the conflict-free chromatic guarding of polygons, for which we have described a polynomial-time algorithm that uses $O(n \log^2 n)$ colors to guard a polygon in a conflict-free setting, and proper coloring of polygon visibility graphs, for which we have described an algorithm that returns a proper 4-coloring for a simple polygon. Besides, we have shown that the 5-colorability problem is NP-complete on visibilit…
