Related papers: Majority Digraphs

A reflexive cycle is any reflexive digraph whose underlying undirected graph is a cycle. Call a relational structure Slupecki if its surjective polymorphisms are all essentially unary. We prove that all reflexive cycles of girth at least 4…

For any graph G = (V, E) and proportion $p\in(0,1]$, a set $S\subseteq V$ is a p-dominating set if $\frac{|N[S]|}{|V|}\geq p$. The $p$-domination number $\gamma_{p}(G)$ equals the minimum cardinality of a $p$-dominating set in G. For a…

We prove that for every digraph $D$ and every choice of positive integers $k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together with a surjective acyclic homomorphism $\psi\colon D^*\to D$ such that: (i) for every…

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…

Given a graph G=(V, E), a vertex is said to ve-dominate an edge if it is either incident with the edge or adjacent to one of its endpoints. A set of vertices is a ve-dominating set if it ve-dominates every edge of the graph. We introduce…

We view hyper-graphs as incidence graphs, i.e. bipartite graphs with a set of nodes representing vertices and a set of nodes representing hyper-edges, with two nodes being adjacent if the corresponding vertex belongs to the corresponding…

An out-branching $B^+_u$ (in-branching $B^-_u$) in a digraph $D$ is a connected spanning subdigraph of $D$ in which every vertex except the vertex $u$, called the root, has in-degree (out-degree) one. A {\bf good $\mathbf{(u,v)}$-pair} in…

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if…

Given a simple graph $G$ on $n$ vertices, a subset of vertices $U \subseteq V(G)$ is dominating if every vertex of $V(G)$ is either in $U$ or adjacent to a vertex of $U$. The domination polynomial of $G$ is the generating function whose…

Given a graph G equals (V,E), a subset S subset of V is a dominating set if every vertex in V minus S is adjacent to some vertex in S. The dominating set with the least cardinality, gamma, is called a gamma-set which is commonly known as a…

Given a digraph $D$, we denote by $\vec{\alpha}(D)$ the maximum size of an acyclic set of $D$ (i.e. a set of vertices which induces a subdigraph with no directed cycles), and by $\vec\chi(D)$ the minimum number of acyclic sets into which…

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have…

A total dominating set of a graph $G=(V,E)$ is a subset $D$ of $V$ such that every vertex in $V$ is adjacent to at least one vertex in $D$. The total domination number of $G$, denoted by $\gamma _t (G)$, is the minimum cardinality of a…

Let C and D be digraphs. A mapping $f:V(D)\to V(C)$ is a C-colouring if for every arc $uv$ of D, either $f(u)f(v)$ is an arc of C or $f(u)=f(v)$, and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is…

A dominating set $D$ in a digraph is a set of vertices such that every vertex is either in $D$ or has an in-neighbour in $D$. A dominating set $D$ of a digraph is locating-dominating if every vertex not in $D$ has a unique set of…

Suppose $D = (V, E)$ is a strongly connected digraph and $u, v \in V (D)$. Among the many metrics in graphs, the sum metric warrants further exploration. The sum distance $sd(u, v)$ defined as $sd(u, v) =\overrightarrow{d}(u,…

The Friendship Theorem states that if in a party any pair of persons has precisely one common friend, then there is always a person who is everybody's friend and the theorem has been proved by Paul Erd\H{o}s, Alfr\'{e}d R\'{e}nyi, and Vera…

A graph is an opposition graph, respectively, a coalition graph, if it admits an acyclic orientation which puts the two end-edges of every chordless 4-vertex path in opposition, respectively, in the same direction. Opposition and coalition…

We define, for any graph $G=(V,E)$, a boundary $\partial G \subseteq V$. The definition coincides with what one would expected for the discretization of (sufficiently nice) Euclidean domains and contains all vertices from the…

For a digraph $D$ of order $n$ and an integer $1 \leq k \leq n-1$, the $k$-token digraph of $D$ is the graph whose vertices are all $k$-subsets of vertices of $D$ and, given two such $k$-subsets $A$ and $B$, $(A,B)$ is an arc in the…