A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain conditions. Graph labelings were first introduced in the late 1960s. In the intervening years dozens of graph labelings techniques have been studied in over 1000 papers. Finding out what has been done for any particular kind of labeling and keeping up with new discoveries is difficult because of the sheer number of papers and because many of the papers have appeared in journals that are not widely available. In this survey I have collected everything I could find on graph labeling. For the convenience of the reader the survey includes a detailed table of contents and index.

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A Dynamic Survey of Graph Labeling

A graph is just a bunch of points with lines between some of them, like a map of cities linked by roads. A rather simple notion. Nevertheless, the theory of graphs has broad and important applications, because so many things can be modeled by graphs. For example, planar graphs — graphs in which none of the lines cross are— important in designing computer chips and other electronic circuits. Also, various puzzles and games are solved easily if a little graph theory is applied.

A Theory of Graphs

A set of vertices $S$ resolves a graph $G$ 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 $G\,\square\,H$. We prove that the metric dimension of $G\,\square\,G$ 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 $G\,\square\,H$ 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 $G\,\square\,G$ is unbounded.

https://users.monash.edu.au/~davidwo/papers/MetricDimension-arXiv.pdf

On the Metric Dimension of Cartesian Products of Graphs

We study generators of metric spaces--sets of points with the property that every point of the space is uniquely determined by the distances from their elements. Such generators put a light on seemingly different kinds of problems in combinatorics that are not directly related to metric spaces. The two applications we present concern combinatorial search: problems on false coins known from the borderline of extremal combinatorics and information theory; and a problem known from combinatorial optimization--connected joins in graphs.We use results on the detection of false coins to approximate the metric dimension (minimum size of a generator for the metric space defined by the distances) of some particular graphs for which the problem was known and open. In the opposite direction, using metric generators, we show that the existence of connected joins in graphs can be solved in polynomial time, a problem asked in a survey paper of Frank. On the negative side we prove that the minimization of the number of components of a join is NP-hard.We further explore the metric dimension with some problems. The main problem we are led to is how to extend an isometry given on a metric generator of a metric space.

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On Metric Generators of Graphs

For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element. Therefore, for any given data element or node in the list structure, we can talk in terms of a next element and a previous element.

Graphs and Digraphs

Resolvability in graphs and the metric dimension of a graph

Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the “AIM Minimum Rank–Special Graphs Work Group”, whereas metric dimension is a well-known graph parameter. We investigate the metric dimension and the zero forcing number of some line graphs by first determining the metric dimension and the zero forcing number of the line graphs of wheel graphs and the bouquet of circles. We prove that $Z(G) \le 2Z(L(G))$ for a simple and connected graph $G$. Further, we show that $Z(G) \le Z(L(G))$ when $G$ is a tree or when $G$ contains a Hamiltonian path and has a certain number of edges. We compare the metric dimension with the zero forcing number of a line graph by demonstrating a couple of inequalities between the two parameters. We end by stating some open problems.

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Metric dimension and zero forcing number of two families of line graphs

The metric dimension dim(G) of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z(G) of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in V(G)S are colored white) such that V(G) is turned black after finitely many applications of “the color-change rule”: a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim(T) ≤ Z(T) for a tree T, and that dim(G) ≤ Z(G)+1 if G is a unicyclic graph; along the way, we characterize trees T attaining dim(T) = Z(T). For a general graph G, we introduce the “cycle rank conjecture”. We conclude with a proof of dim(T) − 2 ≤ dim(T + e) ≤ dim(T) + 1 for $$e \in E\left( {\bar T} \right)$$
 .

A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs

The metric dimension dim(G) of a graph G is the minimum cardinality of a set of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. Let v and e respectively denote a vertex and an edge of a graph G. We show that, for any integer k, there exists a graph G such that dim(G − v) − dim(G) = k. For an arbitrary edge e of any graph G, we prove that dim(G − e) ≤ dim(G) + 2. We also prove that dim(G − e) ≥ dim(G) − 1 for G belonging to a rather general class of graphs. Moreover, we give an example showing that dim(G)− dim(G− e) can be arbitrarily large.

https://www.intlpress.com/site/pub/files/_fulltext/journals/joc/2015/0006/0004/JOC-2015-0006-0004-a002.pdf

The effect of vertex or edge deletion on the metric dimension of graphs

Let G be a connected graph with vertex set V (G) and edge set E(G). A (defensive) alliance in G is a subset S of V (G) such that for every vertex v ∈ S, |N [v]∩S| ≥ |N(v)∩(V (G)− S)|. The alliance partition number, ψa(G), was defined (and further studied in [11]) to be the maximum number of sets in a partition of V (G) such that each set is a (defensive) alliance. Similarly, ψg(G) is the maximum number of sets in a partition of V (G) such that each set is a global alliance, i.e. each set is an alliance and a dominating set. In this paper, we give bounds for the global alliance partition number in terms of the minimum degree, which gives exactly two values for ψg(G) in trees. We concentrate on conditions that classify trees to have ψg(G) = i (i = 1, 2), presenting a characterization for binary trees.

/pdf/global-alliance-partition-in-trees-jriv228xmu.pdf

Linda Eroh

Papers

Resolvability in graphs and the metric dimension of a graph

Metric dimension and zero forcing number of two families of line graphs

A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs

The effect of vertex or edge deletion on the metric dimension of graphs

Global Alliance Partition in Trees