Related papers: Optimal bounds on codes for location in circulant …
A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A restrained dominating set of $G$ is a dominating set $S$ with the additional restraint that the graph $G…
Let $G$ be the circulant graph $C_n(S)$ with $S \subseteq \{1, 2, \dots, \lfloor \frac{n}{2} \rfloor\}$, and let $I(G)$ denote the edge ideal in the polynomial ring $R=\mathbb{K}[x_0, x_1, \dots, x_{n-1}]$ over a field $\mathbb{K}$. In this…
The aim of this work is to investigate the nonnegative signed domination number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and prove…
In this paper we first analyze the behaviour of the identifying code number under union and join operations in graphs. This study forced us to analyze three new parameters related to identifying codes, dominating sets and total dominating…
In $2006$, Danielsen and Parker \cite{DP} proved that every self-dual additive code over $GF(4)$ is equivalent to a graph code. So, graph is an important tool for searching (proposed) optimum codes. In this paper, we introduce a new method…
A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ that is an independent set such that every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$ of $V$ such…
A set $D$ of vertices in a graph $G = (V, E)$ is a locating-dominating set (LD-set) if it is dominating and every two vertices $u$, $v$ of $V\setminus D$ satisfy $N(u) \cap D \neq N(v) \cap D$. Two disjoint sets $A,B\subset V(G)$ form a…
We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V|)+1 where V is the set of vertices of the graph.…
In this paper, we study the signed domination numbers of graphs and present new sharp lower and upper bounds for this parameter. As an example, we present a lower bound on signed domination number of trees in terms of the order, leaves and…
An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum…
We prove that for every integer $d \ge 3$, the median eigenvalues of any graph of maximum degree $d$ are bounded above by $\sqrt{d-1}$. We also prove that, in three separate cases, the median eigenvalues of a graph of maximum degree $d$ are…
Codes over trees were introduced recently to bridge graph theory and coding theory with diverse applications in computer science and beyond. A central challenge lies in determining the maximum number of labelled trees over $n$ nodes with…
This paper deals with the maximum value of the difference between the determining number and the metric dimension of a graph as a function of its order. Our technique requires to use locating-dominating sets, and perform an independent…
Codes defined on graphs and their properties have been subjects of intense recent research. On the practical side, constructions for capacity-approaching codes are graphical. On the theoretical side, codes on graphs provide several…
A dominating set in a directed graph is a set of vertices $S$ such that all the vertices that do not belong to $S$ have an in-neighbour in $S$. A locating set $S$ is a set of vertices such that all the vertices that do not belong to $S$ are…
We call a subset $C$ of vertices of a graph $G$ a $(1,\leq \ell)$-identifying code if for all subsets $X$ of vertices with size at most $\ell$, the sets $\{c\in C |\exists u \in X, d(u,c)\leq 1\}$ are distinct. The concept of identifying…
An identifying code in a graph is a set of vertices which intersects all the symmetric differences between pairs of neighbourhoods of vertices. Not all graphs have identifying codes; those that do are referred to as twin-free. In this…
In this short note, we prove that every twin-free graph on $n$ vertices contains a locating-dominating set of size at most $\lceil\frac{5}{8}n\rceil$. This improves the earlier bound of $\lfloor\frac{2}{3}n\rfloor$ due to Foucaud, Henning,…
A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized,…
We study the existence and the number of $k$-dominating independent sets in certain graph families. While the case $k=1$ namely the case of maximal independent sets - which is originated from Erd\H{o}s and Moser - is widely investigated,…