Related papers: Assorted Musings on Dimension-critical Graphs

The metric dimension of a graph $G$ is the size of a smallest subset $L \subseteq V(G)$ such that for any $x,y \in V(G)$ with $x\not= y$ there is a $z \in L$ such that the graph distance between $x$ and $z$ differs from the graph distance…

A divisor graph $G$ is an ordered pair $(V, E)$ where $V \subset \mathbbm{Z}$ and for all $u \neq v \in V$, $u v \in E$ if and only if $u \mid v$ or $v \mid u$. A graph which is isomorphic to a divisor graph is also called a divisor graph.…

We introduce a variation of metric dimension, called the multiset dimension. The representation multiset of a vertex $v$ with respect to $W$ (which is a subset of the vertex set of a graph $G$), $r_m (v|W)$, is defined as a multiset of…

Given a connected graph $G(V, E)$, the edge dimension, denoted $\mathrm{edim}(G)$, is the least size of a set $S \subseteq V$ that distinguishes every pair of edges of $G$, in the sense that the edges have pairwise distinct tuples of…

Let G be a simple finite graph such that each vertex has an integer value and different vertices have different values. Let S be a finite non-empty set of primes. We call G an S-graph if any two vertices are connected by an edge if and only…

A near-factor of a finite simple graph $G$ is a matching that saturates all vertices except one. A graph $G$ is said to be near-factor-critical if the deletion of any vertex from $G$ results in a subgraph that has a near-factor. We prove…

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars…

Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut dimension of $G$ is the dimension of the span of the characteristic vectors of the minimum cuts of $G$, viewed as vectors in $\{0,1\}^m$. For every $n \ge 2$ we show…

Erd\H{o}s, Harary, and Tutte defined the dimension of a graph $G$ as the smallest natural number $n$ such that $G$ can be embedded in $\mathbb{R}^n$ with each edge a straight line segment of length 1. Since the proposal of this definition,…

Let $G$ be a (di)graph. A set $W$ of vertices in $G$ is a \emph{resolving set} of $G$ if every vertex $u$ of $G$ is uniquely determined by its vector of distances to all the vertices in $W$. The \emph{metric dimension} $\mu (G)$ of $G$ is…

A subset $D$ of $V$ is \emph{dominating} in $G$ if every vertex of $V-D$ has at least one neighbour in $D;$ let $\gamma(G)$ be the minimum cardinality among all dominating sets in $G.$ A graph $G$ is $\gamma$-$q$-{\it critical} if the…

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. We say that a graph $G$ is $d$-distinguishing critical, if…

In this paper, we introduce a connection between two classical concepts of graph theory: \; metric dimension and distinguishing number. For a given graph $G$, let ${\rm dim}(G)$ and $D(G)$ represent its metric dimension and distinguishing…

The \emph{distance-number} of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the…

A vertex set $U \subseteq V$ of an undirected graph $G=(V,E)$ is a $\textit{resolving set}$ for $G$, if for every two distinct vertices $u,v \in V$ there is a vertex $w \in U$ such that the distances between $u$ and $w$ and the distance…

Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent, alpha(G) is the size of a maximum independent set, and core(G) is the intersection of all maximum independent sets. The number d(X)=|X|-|N(X)| is the…

A set $S\subseteq V$ is \textit{independent} in a graph $G=\left( V,E\right) $ if no two vertices from $S$ are adjacent. The \textit{independence number} $\alpha(G)$ is the cardinality of a maximum independent set, while $\mu(G)$ is the…

The algebraic degree $Deg(G)$ of a graph $G$ is the dimension of the splitting field of the adjacency polynomial of $G$ over the field $\mathbb{Q}$. It can be shown that for every positive integer $d$, there exists a circulant graph with…

Given a simple graph $G$, denote by $\Delta(G)$, $\delta(G)$, and $\chi'(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is \emph{$\Delta$-critical} if $\chi'(G)=\Delta(G)+1$ and…

For a finite group $G$, let $\Delta(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we show that if the diameter of $\Delta(G)$ is equal to three, then the complement…