Eunjeong Yi

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)

ON METRIC DIMENSION OF FUNCTIGRAPHS

e∈E(T¯).

On metric dimension of permutation graphs

Zero forcing number, Z(G), of a graph G is the minimum cardinality of a set S of black vertices (whereas vertices in \(V(G)\!\setminus\!S\) 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. Zero forcing number was introduced and used to bound the minimum rank of graphs by the “AIM Minimum Rank – Special Graphs Work Group”. Let G 1 and G 2 be disjoint copies of a graph G and let σ: V(G 1) → V(G 2) be a permutation. Then a permutation graph G σ = (V, E) has the vertex set V = V(G 1) ∪ V(G 2) and the edge set E = E(G 1) ∪ E(G 2) ∪ {uv |v = σ(u)}. It is readily seen that 1 + δ(G) ≤ Z(G σ ) ≤ n, if G is a graph of order n ≥ 2; here δ(G) is the minimum degree of G. We give examples showing that |Z(G) − Z(G σ )| can be arbitrarily large. Further, we characterize permutation graphs G σ satisfying Z(G σ ) = n for a graph G that is a nearly complete graph, a complete k-partite graph, a cycle, and a path, respectively, on n vertices.

The fractional strong metric dimension in three graph products

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.

A Comparison Between the Zero Forcing Number and the Strong Metric Dimension of Graphs

For any two vertices x and y of a graph G, let S{x, y} denote the set of vertices z such that either x lies on a y − z geodesic or y lies on a x − z geodesic. For a function g defined on V(G) and U ⊆ V(G), let g(U) = ∑ x ∈ Ug(x). A function g: V(G) → [0,1] is a strong resolving function of G if g(S{x, y}) ≥ 1, for every pair of distinct vertices x, y of G. The fractional strong metric dimension, sdim f (G), of a graph G is min {g(V(G)): g is a strong resolving function of G}. For any connected graph G of order n ≥ 2, we prove the sharp bounds \(1 \le sdim_f(G) \le \frac{n}{2}\). Indeed, we show that sdim f (G) = 1 if and only if G is a path. If G contains a cut-vertex, then \(sdim_f(G) \le \frac{n-1}{2}\) is the sharp bound. We determine sdim f (G) when G is a tree, a cycle, a wheel, a complete k-partite graph, or the Petersen graph. For any tree T, we prove the sharp inequality sdim f (T + e) ≥ sdim f (T) and show that sdim f (G + e) − sdim f (G) can be arbitrarily large. Lastly, we furnish a Nordhaus-Gaddum-type result: Let G and \(\overline{G}\) (the complement of G) both be connected graphs of order n ≥ 4; it is readily seen that \(sdim_f(G)+sdim_f(\overline{G})=2\) if and only if n = 4; further, we characterize unicyclic graphs G attaining \(sdim_f(G)+sdim_f(\overline{G})=n\).

Disjunctive total domination in permutation graphs

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.

Iteration Index of a Zero Forcing Set in a Graph

The metric dimension