Related papers: Domination numbers and noncover complexes of hyper…
Let $G=(V,E)$ be a simple graph. A set $D\subseteq V$ is a strong dominating set of $G$, if for every vertex $x\in V\setminus D$ there is a vertex $y\in D$ with $xy\in E(G)$ and $deg(x)\leq deg(y)$. The strong domination number…
The article starts with some introductory material about resolution graphs of normal surface singularities (definitions, topological/homological properties, etc). We then discuss the case when the normal surface singularity is an N-fold…
For a class $\mathcal{H}$ of graphs, #Sub$(\mathcal{H})$ is the counting problem that, given a graph $H\in \mathcal{H}$ and an arbitrary graph $G$, asks for the number of subgraphs of $G$ isomorphic to $H$. It is known that if $\mathcal{H}$…
Graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ and $H$ is a subgraph of $G+e$ for any edge $e$ not in $G$. The saturation number for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. In this…
A graph with degree sequence $\pi$ is a \emph{unigraph} if it is isomorphic to every graph that has degree sequence $\pi$. The class of unigraphs is not hereditary and in this paper we study the related hereditary class HCU, the hereditary…
While a number of bounds are known on the zero forcing number $Z(G)$ of a graph $G$ expressed in terms of the order of a graph and maximum or minimum degree, we present two bounds that are related to the (upper) total domination number…
We prove the following result: If $G$ be a connected graph on $n \ge 6$ vertices, then there exists a set of vertices $D$ with $|D| \le \frac{n}{3}$ and such that $V(G) \setminus N[D]$ is an independent set, where $N[D]$ is the closed…
How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given $r$-edge-coloured graph $G$? These problems were introduced in the 1960s and were intensively studied by various researchers over the last…
The {\em independent domination number} $\gamma^i(G)$ of a graph $G$ is the maximum, over all independent sets $I$, of the minimal number of vertices needed to dominate $I$. It is known \cite{abz} that in chordal graphs $\gamma^i$ is equal…
We prove a transfer theorem for hereditary classes of $(r+1)$-uniform hypergraphs. Let $\mathcal H$ be such a class, and for $H\in\mathcal H$ write $\Delta(H)$ and $d(H)$ for the maximum degree and average degree of $H$, respectively. We…
In this paper we develop a framework to study observability for uniform hypergraphs. Hypergraphs, being extensions of graphs, allow edges to connect multiple nodes and unambiguously represent multi-way relationships which are ubiquitous in…
In a graph $G$, a vertex dominates itself and its neighbours. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The minimum cardinality among all double dominating…
In this text we expose basic cases of some fundamental ideas and methods of topology. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal…
Let $H$ be a fixed graph. We say that a graph $G$ is $H$-saturated if it has no subgraph isomorphic to $H$, but the addition of any edge to $G$ results in an $H$-subgraph. The saturation number $\mathrm{sat}(H,n)$ is the minimum number of…
In this paper we study combinatorial and algorithmic resp. complexity questions of upper domination, i.e., the maximum cardinality of a minimal dominating set in a graph. We give a full classification of the related maximisation and…
We study the relationship between the eternal domination number of a graph and its clique covering number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt. We…
For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating…
The notion of graph covers is a discretization of covering spaces introduced and deeply studied in topology. In discrete mathematics and theoretical computer science, they have attained a lot of attention from both the structural and…
The independence complex of a graph is a simplicial complex whose faces correspond to the independent sets of $G$. While independence complexes have been studied extensively for many graph classes, including square grid graphs, relatively…
We compute the homology of the matching complex $M(\Gamma)$, where $\Gamma$ is the complete hypergraph on $n\geq 2$ vertices, and analyse the $S_n$-representations carried by this homology. These results are achieved using standard…