Linda Eroh

Resolvability in graphs and the metric dimension of a graph

Abstract For an ordered subset W={w1,w2,…,wk} of vertices in a connected graph G and a vertex v of G, the metric representation of v with respect to W is the k-vector r(v | W)=(d(v,w 1 ) , d(v,w2),…,d(v,wk)). The set W is a resolving set for G if r(u | W)=r(v | W) implies that u=v for all pairs u,v of vertices of G. The metric dimension dim(G) of G is the minimum cardinality of a resolving set for G. Bounds on dim(G) are presented in terms of the order and the diameter of G. All connected graphs of order n having dimension 1,n−2, or n−1 are determined. A new proof for the dimension of a tree is also presented. From this result sharp bounds on the metric dimension of unicyclic graphs are established. It is shown that dim(H)⩽dim(H×K2)⩽dim(H)+1 for every connected graph H. Moreover, it is shown that for every positive real number e, there exists a connected graph G and a connected induced subgraph H of G such that dim(G)/dim(H)

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 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

DOMINATION IN FUNCTIGRAPHS

e \in E\left( {\bar T} \right)

γ-labelings of complete bipartite graphs

e∈E(T¯).

Closed k-stop distance in graphs

The matching graph M(G) of a graph G is that graph whose vertices are the maximum matchings in G and where two vertices M1 and M2 of M(G) are adjacent if and only if |M1 - M2| = 1. When M(G) is connected, this graph models a metric space whose metric is defined on the set of maximum matchings in G. Which graphs are matching graphs of some graph is not known in general. We determine several forbidden induced subgraphs of matching graphs and add even cycles to the list of known matching graphs. In another direction, we study the behavior of sequences of iterated matching graphs.

A Graph Theoretical Analysis of the Number of Edges in k-Dense Graphs

Let G1 and G2 be disjoint copies of a graph G, and let f : V (G1) → V (G2) be a function. Then a functigraph C(G,f) = (V,E) has the vertex set V = V (G1) ∪ V (G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | u ∈ V (G1),v ∈ V (G2),v = f(u)}. A functigraph is a generalization of a permutation graph (also known as a generalized prism) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let γ(G) denote the domination number of G. It is readily seen that γ(G) ≤ γ(C(G,f)) ≤ 2γ(G). We investigate for graphs generally, and for cycles in great detail, the functions which achieve the upper and lower bounds, as well as the realization of the intermediate values.

Knight’s Tours on Rectangular Chessboards Using External Squares

The metric dimension of a graph G, denoted by dim(G), is the minimum number of vertices such that each vertex is uniquely determined by its distances to the chosen vertices. Let G1 and G2 be disjoint copies of a graph G and let f : V(G1) → V(G2) be a function. Then a functigraphC(G, f) = (V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = f(u)}. We study how metric dimension behaves in passing from G to C(G, f) by first showing that 2 ≤ dim(C(G, f)) ≤ 2n - 3, if G is a connected graph of order n ≥ 3 and f is any function. We further investigate the metric dimension of functigraphs on complete graphs and on cycles.

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

Explicit formulae for the -min and -max labeling values of complete bipartite graphs are given, along with -labelings which achieve these extremes. A recursive formula for the -min labeling value of any complete multipartite is also presented.