Related papers: Lower General Position in Cartesian Products

The coherent configuration $\mathsf{WL}(X)$ of a graph $X$ is the smallest coherent configuration on the vertices of $X$ that contains the edge set of $X$ as a relation. The aim of the paper is to study $\mathsf{WL}(X)$ when $X$ is a…

The \emph{general position problem} in graphs asks for a largest set of vertices in which no three lie on a common shortest path. The \emph{mutual-visibility problem} seeks a largest set of vertices such that every pair is connected by a…

Given a set of points in the plane, the \textsc{General Position Subset Selection} problem is that of finding a maximum-size subset of points in general position, i.e., with no three points collinear. The problem is known to be ${\rm…

The strong geodetic problem on a graph $G$ is to determine a smallest set of vertices such that by fixing one shortest path between each pair of its vertices, all vertices of $G$ are covered. To do this as efficiently as possible, strong…

A set of vertices of a graph is said to be in general position if no three vertices from the set lie on a common geodesic. Recently Klav\v{z}ar, Rall and Yero generalized this notion by defining a set of vertices to be in general…

In this paper, we study the computational complexity of finding the \emph{geodetic number} of graphs. A set of vertices $S$ of a graph $G$ is a \emph{geodetic set} if any vertex of $G$ lies in some shortest path between some pair of…

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…

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…

Let $G=(V,E)$ be a connected graph. The distance between two vertices $u,v\in V$, denoted by $d(u, v)$, is the length of a shortest $u-v$ path in $G$. The distance between a vertex $v\in V$ and a subset $P\subset V$ is defined as $min\{d(v,…

The \emph{generalized sorting problem} is a restricted version of standard comparison sorting where we wish to sort $n$ elements but only a subset of pairs are allowed to be compared. Formally, there is some known graph $G = (V, E)$ on the…

Given a graph, the shortest-path problem requires finding a sequence of edges with minimum cumulative length that connects a source vertex to a target vertex. We consider a variant of this classical problem in which the position of each…

Let $G = (V, E)$ be a simple undirected connected graph. A set $C \subseteq V(G)$ is weakly convex in $G$ if for every two vertices $u,v$ in $G$, there exists a $u-v$ geodesic whose vertices are in $C$. A set $C \subseteq V$ is an…

A vertex $v\in V$ is said to resolve two vertices $x$ and $y$ if $d_G(v,x)\ne d_G(v,y)$. A set $S\subset V$ is said to be a metric generator for $G$ if any pair of vertices of $G$ is resolved by some element of $S$. A minimum metric…

Let $H$, $T$ and $C_n$ be a graph, a tree and a cycle of order $n$, respectively. Let $H^{(i)}$ be the complete join of $H$ and an empty graph on $i$ vertices. Then the Cartesian product $H\Box T$ of $H$ and $T$ can be obtained by applying…

For a connected graph $G$, a vertex subset $S$ of $V(G)$ is a safe set if for every component $C$ of the subgraph of $G$ induced by $S$, $|C| \ge |D|$ holds for every component $D$ of $G-S$ such that there exists an edge between $C$ and…

Let $G$ be a connected graph. A vertex $w$ {\em strongly resolves} a pair $u, v$ of vertices of $G$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $W$ of vertices is a {\em strong…

Given a graph $G$, the (graph theory) general position problem is to find the maximum number of vertices such that no three vertices lie on a common geodesic. This graph invariant is called the general position number (gp-number for short)…

A subset $S \subseteq V$ in a graph $G = (V,E)$ is called a $[1, k]$-set, if for every vertex $v \in V \setminus S$, $1 \leq | N_G(v) \cap S | \leq k$. The $[1,k]$-domination number of $G$, denoted by $\gamma_{[1, k]}(G)$ is the size of the…

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…

A global offensive alliance in a graph $G$ is a set $S$ of vertices with the property that every vertex not belonging to $S$ has at least one more neighbor in $S$ than it has outside of $S$. The global offensive alliance number of $G$,…