Related papers: Strong edge geodetic problem on grids

Let $G=(V,E)$ be a graph. A subset $D$ of $V(G)$ is called a super dominating set if for every $v \in V(G)-D$ there exists an external private neighbour of $v$ with respect to $V(G)-D.$ The minimum cardinality of a super dominating set is…

Given a graph G = (V,E) with a root r in V, positive capacities {c(e)|e in E}, and non-negative lengths {l(e)|e in E}, the minimum-length (rooted) edge capacitated Steiner tree problem is to find a tree in G of minimum total length, rooted…

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric…

Let $G=(V,E)$ be a simple graph. A set $D\subseteq V$ is a strong dominating set of $G$, if for every vertex $x\in V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number…

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…

An identifying code in a graph is a dominating set that also has the property that the closed neighborhood of each vertex in the graph has a distinct intersection with the set. The minimum cardinality of an identifying code, or ID code, in…

A set $D$ of vertices in a graph $G$ is a dominating set if every vertex of $G$, which is not in $D$, has a neighbor in $D$. A set of vertices $D$ in $G$ is convex (respectively, isometric), if all vertices in all shortest paths…

A \textit{maximum stable set} in a graph $G$ is a stable set of maximum cardinality. $S$ is a \textit{local maximum stable set} of $G$, and we write $S\in\Psi(G)$, if $S$ is a maximum stable set of the subgraph induced by $S\cup N(S)$,…

A mixed graph $G$ is a graph that consists of both undirected and directed edges. An orientation of $G$ is formed by orienting all the undirected edges of $G$, i.e., converting each undirected edge $\{u,v\}$ into a directed edge that is…

A path system in a graph $G$ is a collection of paths, with exactly one path between any two vertices in $G$. A path system is said to be consistent if it is intersection-closed. We show that the number of consistent path systems on $n$…

A monitoring edge-geodetic set of a graph is a subset $M$ of its vertices such that for every edge $e$ in the graph, deleting $e$ increases the distance between at least one pair of vertices in $M$. We study the following computational…

The Grundy domination number of a simple graph $G = (V,E)$ is the length of the longest sequence of unique vertices $S = (v_1, \ldots, v_k)$, $v_i \in V$, that satisfies the property $N[v_i] \setminus \cup_{j=1}^{i-1}N[v_j] \neq \emptyset$…

We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all $n$-vertex cubic graphs is attained by a $2$-connected graph $G=(V,E)$. By Petersen's graph…

The toughness of a graph $G$ is defined as the largest real number $t$ such that for any set $S\subseteq V(G)$ such that $G-S$ is disconnected, $S$ has at least $t$ times more elements than $G-S$ has components (unless $G$ is complete, in…

Determining the crossing numbers of Cartesian products of small graphs with arbitrarily large paths has been an ongoing topic of research since the 1970s. Doing so requires the establishment of coincident upper and lower bounds; the former…

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is…

The strong resolving graph $G_{SR}$ of a connected graph $G$ was introduced in [Discrete Applied Mathematics 155 (1) (2007) 356--364] as a tool to study the strong metric dimension of $G$. Basically, it was shown that the problem of finding…

The strong vertex (edge) span of a given graph $G$ is the maximum distance that two players can maintain at all times while visiting all vertices (edges) of $G$ and moving either to an adjacent vertex or staying in the current position…

An edge-coloring of a graph $G$ assigns a color to each edge of $G$. An edge-coloring is a parity edge-coloring if for each path $P$ in $G$, it uses some color on an odd number of edges in $P$. It is a strong parity edge-coloring if for…

Let $G$ be a finite simple graph, and $J_G$ denote the binomial edge ideal of $G$. In this article, we first compute the $\mathrm{v}$-number of binomial edge ideals corresponding to Cohen-Macaulay closed graphs. As a consequence, we obtain…