Related papers: Edge general position sets in Fibonacci and Lucas …

A set of edges $X\subseteq E(G)$ of a graph $G$ is an edge general position set if no three edges from $X$ lie on a common shortest path. The edge general position number ${\rm gp}_{\rm e}(G)$ of $G$ is the cardinality of a largest edge…

Given a graph $G$, the general position problem is to find a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is called a ${\rm gp}$-$set$ of $G$ and its cardinality is the ${\rm…

The general position problem is to find the cardinality of a largest vertex subset S such that no triple of vertices of S lie on a common geodesic. For a connected graph G, the cardinality of S is denoted by gp(G) and called gp-number (or…

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number ${\rm…

The edge geodesic cover problem of a graph $G$ is to find a smallest number of geodesics that cover the edge set of $G$. The edge $k$-general position problem is introduced as the problem to find a largest set $S$ of edges of $G$ such that…

Inspired by a chessboard puzzle of Dudeney, the general position problem in graph theory asks for a largest set $S$ of vertices in a graph such that no three elements of $S$ lie on a common shortest path. The number of vertices in such a…

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number…

Getting inspired by the famous no-three-in-line problem and by the general position subset selection problem from discrete geometry, the same is introduced into graph theory as follows. A set $S$ of vertices in a graph $G$ is a general…

Let $X$ be a vertex subset of a graph $G$. Then $u, v\in V(G)$ are $X$-positionable if $V(P)\cap X \subseteq \{u,v\}$ holds for any shortest $u,v$-path $P$. If each two vertices from $X$ are $X$-positionable, then $X$ is a general position…

If $G$ is a graph, then $X\subseteq V(G)$ is a general position set if for every two vertices $v,u\in X$ and every shortest $(u,v)$-path $P$, it holds that no inner vertex of $P$ lies in $X$. In this note we propose three algorithms to…

A subset $R\subseteq V(G)$ of a graph $G$ is a general position set if any triple set $R_0$ of $R$ is non-geodesic in $G$, that is, no vertex of $R_0$ lies on any geodesic between the other two vertices of $R_0$ in $G$. Let $\mathcal{R}$ be…

The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the…

The general position problem for graphs was inspired by the no-three-in-line problem from discrete geometry. A set $S$ of vertices of a graph $G$ is a \emph{general position set} if no shortest path in $G$ contains three or more vertices of…

A set of vertices $X$ of a graph $G$ is a strong edge geodetic set if to any pair of vertices from $X$ we can assign one (or zero) shortest path between them such that every edge of $G$ is contained in at least one on these paths. The…

The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three distinct vertices from $S$ lie on a common geodesic; such sets are refereed to as gp-sets of $G$. The…

A coalition in a graph $G$ with vertex set $V$ consists of two disjoint sets $V_1, V_2\subset V$ such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1\cup V_2$ is a dominating set in $G$. A partition of graph vertices is…

In a graph $G$, a geodesic between two vertices $x$ and $y$ is a shortest path connecting $x$ to $y$. A subset $S$ of the vertices of $G$ is in general position if no vertex of $S$ lies on any geodesic between two other vertices of $S$. The…

Let $G$ be a graph. A total dominating set of $G$ is a set $S$ of vertices of $G$ such that every vertex is adjacent to at least one vertex in $S$. The total domatic number of a graph is the maximum number of total dominating sets which…

The paper deals with some generalizations of Fibonacci and Lucas sequences, arising from powers of paths and cycles, respectively. In the first part of the work we provide a formula for the number of edges of the Hasse diagram of the…

The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position…