Related papers: Mutual-Visibility of Tree and Its Line Graphs

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…

Let $G=(V(G),E(G))$ be a simple graph, and let $U\subseteq V(G)$. Two distinct vertices $x,y\in U$ are $U$-mutually visible if $G$ contains a shortest $x$-$y$ path that is internally disjoint from $U$. $U$ is called a mutual-visibility set…

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…

The concept of mutual-visibility in graphs has been recently introduced. If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u,…

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 in graphs requires pairs of vertices to be connected by shortest paths that avoid all other vertices of a prescribed set, a condition that is often overly restrictive. In this paper, we introduce a new variant, called…

Let $G$ be a graph and $X\subseteq V(G)$. Then $X$ is a mutual-visibility set if each pair of vertices from $X$ is connected by a geodesic with no internal vertex in $X$. The mutual-visibility number $\mu(G)$ of $G$ is the cardinality of a…

We prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as a subgraph. Our result confirms, for…

A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical…

Let $G$ be a graph and $X\subseteq V(G)$. Then, vertices $x$ and $y$ of $G$ are $X$-visible if there exists a shortest $u,v$-path where no internal vertices belong to $X$. The set $X$ is a mutual-visibility set of $G$ if every two vertices…

An independent edge set of graph $G$ is a matching, and is maximal if it is not a proper subset of any other matching of $G$. The number of all the maximal matchings of $G$ is denoted by $\Psi(G)$. In this paper, an algorithm to count…

The mutual-visibility chromatic number of a graph $G$ is the smallest number of colors needed to color the vertices of $G$ such that each color class is a mutual-visibility set. In this paper, we prove that determining the mutual-visibility…

Mutual visibility in graphs provides a framework for analysing how vertices can observe one another along shortest paths free of internal obstructions. The visibility polynomial, which enumerates mutual-visibility sets of all orders, has…

Given a connected graph $G$, the total mutual-visibility number of $G$, denoted $\mu_t(G)$, is the cardinality of a largest set $S\subseteq V(G)$ such that for every pair of vertices $x,y\in V(G)$ there is a shortest $x,y$-path whose…

Mutual-visibility sets were motivated by visibility in distributed systems and social networks, and intertwine with several classical mathematical areas. Monotone properties of the variety of mutual-visibility sets, and restrictions of such…

If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u,v\}$. If each two vertices from $X$ are $X$-visible, then $X$ is a…

For a given graph \(G\), the general position problem asks for the largest set of vertices \(M \subseteq V(G)\) such that no three distinct vertices of \(M\) belong to a common shortest path in \(G\). A relaxation of this concept is based…

In this paper and a companion paper, we prove that, if $m$ is sufficiently large, every graph on $m+1$ vertices that has a universal vertex and minimum degree at least $\lfloor \frac{2m}{3} \rfloor$ contains each tree $T$ with $m$ edges as…

The general position problem in graphs is to find the maximum number of vertices that can be selected such that no three vertices lie on a common shortest path. The mutual-visibility problem in graphs is to find the maximum number of…

A split-by-edges tree of a graph G on n vertices is a binary tree T where the root = V(G), every leaf is an independent set in G, and for every other node N in T with children L and R there is a pair of vertices {u, v} in N such that L = N…