Related papers: Extremal hypergraphs for matching number and domin…

For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the…

For a graph $G=(V,E)$, a set $S \subseteq V$ is a $[1,2]$-set if it is a dominating set for $G$ and each vertex $v \in V \setminus S$ is dominated by at most two vertices of $S$, i.e. $1 \leq \vert N(v) \cap S \vert \leq 2$. Moreover a set…

Let $\gamma(G)$ denote the domination number of graph $G$. Let $G$ and $H$ be graphs and $G\Box H$ their Cartesian product. For $h\in V(H)$ define $G_h=\{(g,h)\,|\,g\in V(G)\}$ and call this set a $G$-layer of $G\Box H$. We prove the…

As a natural variant of domination in graphs, Dankelmann et al. [Domination with exponential decay, Discrete Math. 309 (2009) 5877-5883] introduce exponential domination, where vertices are considered to have some dominating power that…

Ryser's Conjecture states that any $r$-partite $r$-uniform hypergraph has a vertex cover of size at most $r - 1$ times the size of the largest matching. For $r = 2$, the conjecture is simply K\"onig's Theorem and every bipartite graph is a…

For a graph $H$, the {\em extremal number} $ex(n,H)$ is the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. Let $\delta(H)>0$ and $\Delta(H)$ denote the minimum degree and maximum degree of $H$,…

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

For a maximal outerplanar graph $G$ of order $n$ at least $3$, Matheson and Tarjan showed that $G$ has domination number at most $n/3$. Similarly, for a maximal outerplanar graph $G$ of order $n$ at least $5$, Dorfling, Hattingh, and Jonck…

Let G be a simple undirected graph with no isolated vertex. A paired dominating set of G is a dominating set which induces a subgraph that has a perfect matching. The paired domination number of G, denoted by {\gamma}pr(G), is the size of…

A subset $D$ of vertices of a graph $G$ is a dominating set if for each $u\in V(G)\setminus D$, $u$ is adjacent to some vertex $v\in D$. The domination number, $\gamma(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. For…

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of…

The minimum status of a graph is the minimum of statuses of all vertices of this graph. We give a sharp upper bound for the minimum status of a connected graph with fixed order and matching number (domination number, respectively), and…

We initiate a study of the homomorphism domination exponent of a pair of graphs F and G, defined as the maximum real number c such that |Hom(F,T)| \geq |Hom(G,T)|^c for every graph T. The problem of determining whether HDE(F,G) \geq 1 is…

Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ is a dominating set if every vertex not in $D$ is adjacent to a vertex in $D$. A dominating set $D$ is called a total dominating set if every vertex in $D$ is adjacent to a vertex in $D$.…

One of the foundational theorems of extremal graph theory is Dirac's theorem, which says that if an n-vertex graph G has minimum degree at least n/2, then G has a Hamilton cycle, and therefore a perfect matching (if n is even). Later work…

In this paper, we prove the conjectures of Gharakhloo and Welker (2023) that the positive matching decomposition number (pmd) of a $3$-uniform hypergraph is bounded from above by a polynomial of degree $2$ in terms of the number of…

A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$ vertices in…

Let $G=(V,E)$ be a graph without isolated vertices. A set $S\subseteq V$ is a paired-domination set if every vertex in $V-S$ is adjacent to a vertex in $S$ and the subgraph induced by $S$ contains a perfect matching. The paired-domination…

In this short paper, we establish relations between the domination number $\gamma$, the total domination number $\gamma_t$, and the connected domination number $\gamma_c$ of a graph. In particular, we prove upper and lower bounds for…

For $k \geq 1$, in a graph $G=(V,E)$, a set of vertices $D$ is a distance $k$-dominating set of $G$, if any vertex in $V\setminus D$ is at distance at most $k$ from some vertex in $D$. The minimum cardinality of a distance $k$-dominating…