Related papers: Domination in 3-tournaments

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…

A vertex u of a graph t-dominates a vertex v if there are at most t vertices different from u,v that are adjacent to v and not to u; and a graph is t-dominating if for every pair of distinct vertices, one of them t-dominates the other. Our…

A set of edges $F$ in a graph $G$ is an edge dominating set if every edge in $G$ is either in $F$ or shares a vertex with an edge in $F$. $G$ is said to be well-edge-dominated if all of its minimal edge dominating sets have the same…

A hypergraph is a generalization of a graph where edges can connect any number of vertices. In this paper, we extend the study of locating-dominating sets to hypergraphs. Along with some basic results, sharp bounds for the…

A subset $S$ of vertices in a graph $G$ is a secure total dominating set of $G$ if $S$ is a total dominating set of $G$ and, for each vertex $u \not\in S$, there is a vertex $v \in S$ such that $uv$ is an edge and $(S \setminus \{v\}) \cup…

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…

It is conjectured that the game domination number is at most $3n/5$ for every $n$-vertex graph which does not contain isolated vertices. It was proved in the recent years that the conjecture holds for several graph classes, including the…

For a undirected simple graph $G$, let $d_i(G)$ be the number of $i$-element dominating vertex set of $G$. The domination polynomial of the graph $G$ is defined as $$D(G, x) = \sum_{i = 1}^n d_i(G)x^i.$$ Alikhani and Peng conjectured that…

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…

Motivated by the success of domination games and by a variation of the coloring game called the indicated coloring game, we introduce a version of domination games called the indicated domination game. It is played on an arbitrary graph $G$…

For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $\gamma(G)$ and…

A tournament is a directed graph resulting from an orientation of the complete graph; so, if $M$ is a tournament's adjacency matrix, then $M + M^T$ is a matrix with $0$s on its diagonal and all other entries equal to $1$. An outstanding…

An eternal dominating set of a graph $G$ is a set of vertices (or "guards") which dominates $G$ and which can defend any infinite series of vertex attacks, where an attack is defended by moving one guard along an edge from its current…

A vertex subset $S$ of a graph $G$ is a dominating set if every vertex of $G$ either belongs to $S$ or is adjacent to a vertex of $S$. The cardinality of a smallest dominating set is called the dominating number of $G$ and is denoted by…

Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E' of E which dominates all edges of G. In particular, if every edge of G is dominated by…

A ternary graph is a graph with no induced cycles of length $0$ modulo $3$. It was recently shown that, if the independence complex of a ternary graph is not contractible, then it is homotopy equivalent to a sphere. When a ternary graph…

Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite…

A cycle in a graph is called dominating if every edge of the graph is incident with a vertex of the cycle. In this paper, we investigate forbidden pairs guaranteeing the existence of a dominating cycle in 2-connected graphs.

A dominating set of a graph $G$ is a subset $D$ of vertices such that every vertex not in $D$ is adjacent to at least one vertex in $D$. A dominating set $D$ is paired if the subgraph induced by its vertices has a perfect matching, and…

A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A locating-total dominating set of $G$ is a total dominating set $D$ of $G$ with the additional property that every…