Related papers: Mutual Visibility in Graphs

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…

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,…

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…

Visibility problems have been investigated for a long time under different assumptions as they pose challenging combinatorial problems and are connected to robot navigation problems. The mutual-visibility problem in a graph $G$ of $n$…

Let G(V,E) be a simple graph and let X subset of V. Two vertices u and v are said to be X-visible if there exists a shortest u,v-path P such that V(P) intersection X is a subset of {u, v}. A set X is called a mutual-visibility set of G if…

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…

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…

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…

The concept of mutual visibility in graphs, introduced recently, addresses a fundamental problem in Graph Theory concerning the identification of the largest set of vertices in a graph such that any two vertices have a shortest path…

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…

The mutual-visibility problem in a graph $G$ asks for the cardinality of a largest set of vertices $S\subseteq V(G)$ so that for any two vertices $x,y\in S$ there is a shortest $x,y$-path $P$ so that all internal vertices of $P$ are not in…

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…

Given a graph $G$, a set $X$ of vertices in $G$ satisfying that between every two vertices in $X$ (respectively, in $G$) there is a shortest path whose internal vertices are not in $X$ is a mutual-visibility (respectively, total…

For a connected graph $G$ and $X\subseteq V(G)$, we say that two vertices $u$, $v$ are $X$-visible if there is a shortest $u,v$-path $P$ with $V(P)\cap X \subseteq \{u,v\}$. If every two vertices from $X$ are $X$-visible, then $X$ is a…

For a given graph $G$, the mutual-visibility problem asks for the largest set of vertices $M \subseteq V(G)$ with the property that for any pair of vertices $u,v \in M$ there exists a shortest $u,v$-path of $G$ that does not pass through…

Let $G$ be a graph and $M \subseteq V(G)$. Vertices $x, y \in M$ are $M$-visible if there exists a shortest $x,y$-path of $G$ that does not pass through any vertex of $M \setminus \{x, y \}$. We say that $M$ is a mutual-visibility set if…

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…

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…

Let $G$ be a connected graph and $\cal X \subseteq V(G)$. By definition, two vertices $u$ and $v$ are $\cal X$-visible in $G$ if there exists a shortest $u,v$-path with all internal vertices being outside of the set $\cal X$. The largest…

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…