Related papers: A study on token digraphs

In this paper we enumerate $k$-noncrossing tangled-diagrams. A tangled-diagram is a labeled graph whose vertices are $1,...,n$ have degree $\le 2$, and are arranged in increasing order in a horizontal line. Its arcs are drawn in the upper…

The generalized Kneser hypergraph $KG^{r}(n,k,s)$ is the hypergraph whose vertices are all the $k$-subsets of $\{1,\ldots ,n\}$, and edges are $r$-tuples of distinct vertices such that any pair of them has at most $s$ elements in their…

A graph $G$ is a $D\!D_2$-graph if it has a pair $(D,D_2)$ of disjoint sets of vertices of $G$ such that $D$ is a dominating set and $D_2$ is a 2-dominating set of $G$. We provide several characterizations and hardness results concerning…

Let $G=(V,E)$ be a simple graph. A set $I\subseteq V$ is an independent set, if no two of its members are adjacent in $G$. The $k$-independent graph of $G$, $I_k (G)$, is defined to be the graph whose vertices correspond to the independent…

A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of…

The $k$-core of a graph is its largest subgraph with minimum degree at least $k$, a fundamental concept for uncovering hierarchical structures. In this paper, we establish a connection between the $k$-core and the high-order spectra of…

In this article we investigate the structure of uniformly $k$-connected and uniformly $k$-edge-connected graphs. Whereas both types have previously been studied independent of each other, we analyze relations between these two classes. We…

A topological graph is \emph{$k$-quasi-planar} if it does not contain $k$ pairwise crossing edges. A topological graph is \emph{simple} if every pair of its edges intersect at most once (either at a vertex or at their intersection). In…

The Kneser graph $K(n, k)$ has as vertices all $k$-element subsets of $[n]=\{1,2,...,n \}$ and an edge between any two vertices that are disjoint. If $n=2k+1$, then $K(n, k)$ is called an odd graph. Let $ n >4$ and $1< k < \frac{n}{2} $. In…

A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two of which can be separated in $G$ by fewer than $k$ other vertices. The block number $\beta(G)$ of $G$ is the largest integer $k$ such that $G$ has a $k$-block. We…

A digraph $D=(V,A)$ of order $n\geq 3$ is pancyclic, whenever $D$ contains a directed cycle of length $k$ for each $k\in\{3,...,n\}$; and D is vertex-pancyclic iff, for each vertex $v\in V$ and each $k\in \{3,...,n\}$, $D$ contains a…

The concept of $k$-planarity is extensively studied in the context of Beyond Planarity. A graph is $k$-planar if it admits a drawing in the plane in which each edge is crossed at most $k$ times. The local crossing number of a graph is the…

Given a directed graph G=(V,E) an independent set A of the vertices V is called quasi-kernel (quasi-sink) iff for each point v there is a path of length at most 2 from some point of A to v (from v to some point of A). Every finite directed…

Most state-of-the-art graph kernels only take local graph properties into account, i.e., the kernel is computed with regard to properties of the neighborhood of vertices or other small substructures. On the other hand, kernels that do take…

For a graph $H$ and an integer $k\ge 1$, the \emph{Token Sliding reconfiguration graph} $\mathsf{TS}_k(H)$ and the \emph{Token Jumping reconfiguration graph} $\mathsf{TJ}_k(H)$ have as vertices the $k$-cliques of $H$, with two vertices…

Let $n$, $k$ and $l$ be integers with $1\leq k<l\leq n-1$. The set-inclusion graph $G(n,k,l)$ is the graph whose vertex set consists of all $k$- and $l$-subsets of $[n]=\{1,2,\ldots,n\}$, where two distinct vertices are adjacent if one of…

A kernel in a digraph is an independent and absorbent subset of its vertex set. A digraph is critical kernel imperfect if it does not have a kernel, but every proper induced subdigraph does. In this article, we characterize asymmetrical…

The dichromatic number of a digraph $D$ is the smallest $k$ such that $D$ can be partitioned into $k$ acyclic subdigraphs, and the dichromatic number of an undirected graph is the maximum dichromatic number over all its orientations.…

In a reconfiguration setting, each clique of a graph $G$ is viewed as a set of tokens placed on vertices of $G$ such that no vertex has more than one token and any two tokens are adjacent. Three well-known reconfiguration rules have been…

Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chv\'atal and Lov\'asz proved that…