Related papers: Resolving sets for Johnson and Kneser graphs

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…

A set $R \subseteq V(G)$ is a resolving set of a graph $G$ if for all distinct vertices $v,u \in V(G)$ there exists an element $r \in R$ such that $d(r,v) \neq d(r,u)$. The metric dimension $\dim(G)$ of the graph $G$ is the minimum…

For integers $n\geq k\geq 1$, the Kneser graph $K(n,k)$ is the graph with vertex set $V=[n]^{(k)}$ and edge set $E=\{\{x,y\} \in V^{(2)}: x\cap y=\emptyset\}$. Chen proved that for $n\geq 3k$, Kneser graphs are Hamiltonian and later…

In this paper, the strong and doubly metric dimensions of Johnson and Kneser graphs are considered. The exact value of the strong metric dimension of Johnson graph $J_{n,k}$ is obtained using the well-known results from the literature. The…

Two vertices $u$ and $v$ of an undirected graph $G$ are strongly resolved by a vertex $w$ if there is a shortest path between $w$ and $u$ containing $v$ or a shortest path between $w$ and $v$ containing $u$. A vertex set $R$ is a strong…

Graham and Pollak showed that the vertices of any graph $G$ can be addressed with $N$-tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its…

Let $G=(V,E)$ be a finite, simple, connected, combinatorial graph on $n$ vertices and let $D \in \mathbb{R}^{n \times n}$ be its graph distance matrix $D_{ij} = d(v_i, v_j)$. Steinerberger (J. Graph Theory, 2023) empirically observed that…

We classify the maximal $m$-distance sets in $\mathbb{R}^{n-1}$ which contain the representation of the Johnson graph $J(n, m)$ for $m = 2, 3$. Furthermore, we determine the necessary and sufficient condition for $n$ and $m$ such that the…

Resolving sets were originally designed to locate vertices of a graph one at a time. For the purpose of locating multiple vertices of the graph simultaneously, $\{\ell\}$-resolving sets were recently introduced. In this paper, we present…

For any given $n,m \in \mathbb{N}$ with $ m < n $, the Johnson graph $J(n,m)$ is defined as the graph whose vertex set is $V=\{v\mid v\subseteq [n]=\{1,...,n\}, |v|=m\}$, where two vertices $v$,$w$ are adjacent if and only if $|v\cap…

In this paper, we discuss automorphism related parameters of a graph associated to a finite vector space. The fixing neighborhood of a pair $(u,v)$ of vertices of a graph $G$ is the set of all those vertices $w$ of $G$, such that the orbits…

A subset $Q = \{q_1, q_2, ..., q_l\}$ of vertices of a connected graph $G$ is a doubly resolving set of $G$ if for any various vertices $x, y \in V(G)$ we have $r(x|Q)-r(y|Q)\neq\lambda I$, where $\lambda$ is an integer, and $I$ indicates…

Let $G=(V,E)$ be a connected simple graph. The distance $d(u,v)$ between vertices $u$ and $v$ from $V$ is the number of edges in the shortest $u-v$ path. If $e=uv \in E$ is an edge in $G$ than distance $d(w,e)$ where $w$ is some vertex in…

The Johnson graph J(n,N) is defined as the graph whose vertices are the n-subsets of the set {1,2,...,N}, where two vertices are adjacent if they share exactly n - 1 elements. Unlike Johnson graphs, induced subgraphs of Johnson graphs (JIS…

A vertex $w$ resolves two vertices $u$ and $v$ in a directed graph $G$ if the distance from $w$ to $u$ is different to the distance from $w$ to $v$. A set of vertices $R$ is a resolving set for a directed graph $G$ if for every pair of…

The Kneser graph $K(n,d)$ is the graph on the $d$-subsets of an $n$-set, adjacent when disjoint. Clearly, $K(n+d,d)$ is locally $K(n,d)$. Hall showed for $n \ge 3d+1$ that there are no further examples. Here we give other examples of…

For any non-negative integers $v > k > i$, the {\em generalized Johnson graph}, $J(v,k,i)$, is the undirected simple graph whose vertices are the $k$-subsets of a $v$-set, and where any two vertices $A$ and $B$ are adjacent whenever $|A…

A vertex subset $S$ of a graph $G$ is a general position set of $G$ if no vertex of $S$ lies on a geodesic between two other vertices of $S$. The cardinality of a largest general position set of $G$ is the general position number…

Let $G$ be a connected graph and $\cal X \subseteq V(G)$. By definition, two vertices $u$ and $v$ are $\cal X$-visible in $G$ if there exists a shortest $u,v$-path with all internal vertices being outside of the set $\cal X$. The largest…

A set W \subseteq V (G) is called a resolving set, if for each pair of distinct vertices u,v \in V (G) there exists t \in W such that d(u,t) \neq d(v,t), where d(x,y) is the distance between vertices x and y. The cardinality of a minimum…