Related papers: On the $P_3$-hull number and infecting times of ge…

Let $S$ be an $n$-dimensional vector space over the finite field $\mathbb{F}_q$, where $q$ is necessarily a prime power. Denote $K_q(n,k)$ (resp. $J_q(n,k)$) to be the \emph{$q$-Kneser graph} (resp. \emph{Grassmann graph}) for $k\geq 1$…

The $r$-neighbor bootstrap percolation is a graph infection process based on the update rule by which a vertex with $r$ infected neighbors becomes infected. We say that an initial set of infected vertices propagates if all vertices of a…

The crossing number of a graph is the least number of crossings of edges among all drawings of the graph in the plane. In this article, we prove that the crossing number of the generalized Petersen graph P(10, 3) is equal to 6.

The crossing number of a graph $G$ in a surface $\Sigma$, denoted by $cr_{\Sigma}(G)$, is the minimum number of pairwise intersections of edges in a drawing of $G$ in $\Sigma$. Let $k$ be an integer satisfying $k\geq 3$, the generalized…

Let $G$ be a simple graph and let $S$ be a subset of its vertices. We say that $S$ is $P_3$-convex if every vertex $v \in V(G)$ that has at least two neighbors in $S$ also belongs to $S$. The $P_3$-hull set of $S$ is the smallest…

Let $G=(V(G),E(G))$ be a simple connected and undirected graph with vertex set $V(G)$ and edge set $E(G)$. A set $S \subseteq V(G)$ is a $dominating$ $set$ if for each $v \in V(G)$ either $v \in S$ or $v$ is adjacent to some $w \in S$. That…

A subset $S$ of initially infected vertices of a graph $G$ is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects…

A vertex subset $S$ of a graph $G=(V,E)$ is a $[1,2]$-dominating set if each vertex of $V\backslash S$ is adjacent to either one or two vertices in $S$. The minimum cardinality of a $[1,2]$-dominating set of $G$, denoted by…

Bootstrap percolation on a graph with infection threshold $r\in \mathbb{N}$ is an infection process, which starts from a set of initially infected vertices and in each step every vertex with at least $r$ infected neighbours becomes…

A walk $W$ between vertices $u$ and $v$ of a graph $G$ is called a {\em tolled walk between $u$ and $v$} if $u$, as well as $v$, has exactly one neighbour in $W$. A set $S \subseteq V(G)$ is {\em toll convex} if the vertices contained in…

For $r\geq1$, the $r$-neighbour bootstrap process in a graph $G$ starts with a set of infected vertices and, in each time step, every vertex with at least $r$ infected neighbours becomes infected. The initial infection percolates if every…

Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G, r)$, of…

We present formulas to compute the P3-geodetic number, the P3-hull number and the percolation time for a caterpillar, in terms of certain sequences associated with it. In addition, we find a connection between the percolation time of a unit…

A graph is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersecting disks in G has a common intersection. It is known that for every graph G, there exists a unique smallest Helly graph H(G) into which G…

A set $D$ of vertices of a graph $G$ is isolating if the set of vertices not in $D$ or with no neighbor in $D$ is independent. The isolation number of $G$, denoted by $\iota (G)$, is the minimum cardinality of an isolating set of $G$. It is…

In this paper a new parameter for hypergraphs called hypergraph infection is defined. This concept generalizes zero forcing in graphs to hypergraphs. The exact value of the infection number of complete and complete bipartite hypergraphs is…

We study the following bootstrap percolation process: given a connected graph $G$, a constant $\rho \in [0, 1]$ and an initial set $A \subseteq V(G)$ of \emph{infected} vertices, at each step a vertex~$v$ becomes infected if at least a…

A \textit{primitive hole} of a graph $G$ is a cycle of length $3$ in $G$. The number of primitive holes in a given graph $G$ is called the primitive hole number of that graph $G$. The primitive degree of a vertex $v$ of a given graph $G$ is…

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…

Let $D$ be an orientation of a simple graph. Given $u,v\in V(D)$, a directed shortest $(u,v)$-path is a $(u,v)$-geodesic. $S \subseteq V(D)$ is convex if, for every $u,v \in S$, the vertices in each $(u,v)$-geodesic and in each…