On the metric dimension of corona product graphs

Given a set of vertices $S=\{v_1,v_2,...,v_k\}$ of a connected graph $G$, the metric representation of a vertex $v$ of $G$ with respect to $S$ is the vector $r(v|S)=(d(v,v_1),d(v,v_2),...,d(v,v_k))$, where $d(v,v_i)$, $i\in \{1,...,k\}$ denotes the distance between $v$ and $v_i$. $S$ is a resolving set for $G$ if for every pair of vertices $u,v$ of $G$, $r(u|S)\ne r(v|S)$. The metric dimension of $G$, $dim(G)$, is the minimum cardinality of any resolving set for $G$. Let $G$ and $H$ be two graphs of order $n_1$ and $n_2$, respectively. The corona product $G\odot H$ is defined as the graph obtained from $G$ and $H$ by taking one copy of $G$ and $n_1$ copies of $H$ and joining by an edge each vertex from the $i^{th}$-copy of $H$ with the $i^{th}$-vertex of $G$. For any integer $k\ge 2$, we define the graph $G\odot^k H$ recursively from $G\odot H$ as $G\odot^k H=(G\odot^{k-1} H)\odot H$. We give several results on the metric dimension of $G\odot^k H$. For instance, we show that given two connected graphs $G$ and $H$ of order $n_1\ge 2$ and $n_2\ge 2$, respectively, if the diameter of $H$ is at most two, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(H)$. Moreover, if $n_2\ge 7$ and the diameter of $H$ is greater than five or $H$ is a cycle graph, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(K_1\odot H).

On the partition dimension of trees

Given an ordered partition $\Pi =\{P_1,P_2, ...,P_t\}$ of the vertex set $V$ of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex $v\in V$ with respect to the partition $\Pi$ is the vector $r(v|\Pi)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$ represents the distance between the vertex $v$ and the set $P_i$. A partition $\Pi$ of $V$ is a \emph{resolving partition} of $G$ if different vertices of $G$ have different partition representations, i.e., for every pair of vertices $u,v\in V$, $r(u|\Pi)\ne r(v|\Pi)$. The \emph{partition dimension} of $G$ is the minimum number of sets in any resolving partition of $G$. In this paper we obtain several tight bounds on the partition dimension of trees

On the Metric Dimension of Cartesian Products of Graphs

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded

One size resolvability of graphs

For an ordered set W = {w1,w2, · · · ,wk} of vertices in a connected graph G and a vertex v of G, the code of v with respect to W is the k-vector CW(v) = (d(v,w1), d(v,w2), · · · , d(v,wk)). The set W is a one size resolving set for G if (1) the size of subgraph hWi induced by W is one and (2) distinct vertices of G have distinct code with respect to W. The minimum cardinality of a one size resolving set in graph G is the one size resolving number, denoted by or(G). A one size resolving set of cardinality or(G) is called an or-set of G. We study the existence of or-set in graphs and characterize all nontrivial connected graphs G of order n with or(G) = n and n − 1

Conditional resolvability in graphs: a survey

For an ordered set W={w1,w2,…,wk} of vertices and a vertex v in a connected graph G, the code of v with respect to W is the k-vector cW(v)=(d(v,w1),d(v,w2),…,d(v,wk)), where d(x,y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct codes with respect to W. The minimum cardinality of a resolving set for G is its dimension dim(G). Many resolving parameters are formed by extending resolving sets to different subjects in graph theory, such as the partition of the vertex set, decomposition and coloring in graphs, or by combining resolving property with another graph-theoretic property such as being connected, independent, or acyclic. In this paper, we survey results and open questions on the resolving parameters defined by imposing an additional constraint on resolving sets, resolving partitions, or resolving decompositions in graphs

Connected resolvability of graphs

summary:For an ordered set $W=\lbrace w_1, w_2, \dots , w_k\rbrace$ of vertices and a vertex $v$ in a connected graph $G$, the representation of $v$ with respect to $W$ is the $k$-vector $r(v|W)$ = ($d(v, w_1)$, $d(v, w_2),\dots ,d(v, w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set for $G$ containing a minimum number of vertices is a basis for $G$. The dimension $\dim (G)$ is the number of vertices in a basis for $G$. A resolving set $W$ of $G$ is connected if the subgraph $$ induced by $W$ is a nontrivial connected subgraph of $G$. The minimum cardinality of a connected resolving set in a graph $G$ is its connected resolving number $\mathop {\mathrm cr}(G)$. Thus $1 \le \dim (G) \le \mathop {\mathrm cr}(G) \le n-1$ for every connected graph $G$ of order $n \ge 3$. The connected resolving numbers of some well-known graphs are determined. It is shown that if $G$ is a connected graph of order $n \ge 3$, then $\mathop {\mathrm cr}(G) = n-1$ if and only if $G = K_n$ or $G = K_{1, n-1}$. It is also shown that for positive integers $a$, $b$ with $a \le b$, there exists a connected graph $G$ with $\dim (G) = a$ and $\mathop {\mathrm cr}(G) = b$ if and only if $(a, b) \notin \lbrace (1, k)\: k = 1\hspace{5.0pt}\text{or}\hspace{5.0pt}k \ge 3\rbrace$. Several other realization results are present. The connected resolving numbers of the Cartesian products $G \times K_2$ for connected graphs $G$ are studied

The cardinality of endomorphisms of some oriented paths: an algorithm

An endomorphism of a (oriented) graph is a mapping on the vertex set preserving (arcs) edges. In this paper we provide an algorithm to determine the cardinalities of endomorphism monoids of some ( nite) directed paths, based on results on simple paths.Chiang Mai University; CMUC - Centro de Matemática da Universidade de Coimbra; Srinakharinwirot University; Thailand Research Fund and Commission on Higher Education, Thailand MRG508007

ON RESOLVING EDGE COLORINGS IN GRAPHS

We study the relationships between the resolving edge chromatic number and other graphical parameters and provide bounds for the resolving edge chromatic number of a connected graph

On γ\gamma -labelings of oriented graphs

summary:Let $D$ be an oriented graph of order $n$ and size $m$. A $\gamma$-labeling of $D$ is a one-to-one function $f\: V(D) \rightarrow \lbrace 0, 1, 2, \ldots , m\rbrace$ that induces a labeling $f^{\prime }\: E(D) \rightarrow \lbrace \pm 1, \pm 2, \ldots , \pm m\rbrace$ of the arcs of $D$ defined by $f^{\prime }(e) = f(v)-f(u)$ for each arc $e =(u, v)$ of $D$. The value of a $\gamma$-labeling $f$ is $\mathop {\mathrm val}(f) = \sum _{e \in E(G)} f^{\prime }(e).$ A $\gamma$-labeling of $D$ is balanced if the value of $f$ is 0. An oriented graph $D$ is balanced if $D$ has a balanced labeling. A graph $G$ is orientably balanced if $G$ has a balanced orientation. It is shown that a connected graph $G$ of order $n \ge 2$ is orientably balanced unless $G$ is a tree, n \equiv 2 \hspace{4.44443pt}(\@mod \; 4), and every vertex of $G$ has odd degree

The independent resolving number of a graph

summary:For an ordered set $W=\lbrace w_1, w_2, \dots , w_k\rbrace$ of vertices in a connected graph $G$ and a vertex $v$ of $G$, the code of $v$ with respect to $W$ is the $k$-vector $$c_W(v) = (d(v, w_1), d(v, w_2), \dots , d(v, w_k) ).$$ The set $W$ is an independent resolving set for $G$ if (1) $W$ is independent in $G$ and (2) distinct vertices have distinct codes with respect to $W$. The cardinality of a minimum independent resolving set in $G$ is the independent resolving number $\mathop {\mathrm ir}(G)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order $n$ with $\mathop {\mathrm ir}(G) = 1$, $n-1$, $n-2$, and present several realization results. It is shown that for every pair $r, k$ of integers with $k \ge 2$ and $0 \le r \le k$, there exists a connected graph $G$ with $\mathop {\mathrm ir}(G) = k$ such that exactly $r$ vertices belong to every minimum independent resolving set of $G$