Cong X. Kang

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

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 an x - z geodesic. For a function g defined on V ( G ) and U ź V ( G ) , let g ( U ) = ź x ź U g ( x ) . A function g : V ( G ) ź 0 , 1 is a strong resolving function of G if g ( S { x , y } ) ź 1 , for every pair of distinct vertices x , y of G . The fractional strong metric dimension, s d i m f ( G ) , of a graph G is min { g ( V ( G ) ) : g źisźaźstrongźresolvingźfunctionźofź G } . This paper furthers the study of fractional strong metric dimension initiated in COCOA 2013 (Lecture Notes in Comput. Sci.). First, we clarify or correct the proofs to two characterization theorems contained in two papers on fractional (strong) metric dimension. Next, results on fractional strong metric dimension analogous to the work of Feng, Lv, and Wang on fractional metric dimension are offered. We provide new upper and lower bounds on s d i m f ( G ) , partly in analogy with the work done by Feng et al. and partly by exploiting the particular nature of the strong metric dimension. Finally, motivated by the work of Arumugam, Mathew, and Shen, we describe a class of graphs G for which s d i m f ( G ) = | V ( G ) | 2 .

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.

On domination number and distance in 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 \(sdim_f(G)+sdim_f(\overline{G})=n\).

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

On Lang's Conjecture for Surfaces of General Type

The metric dimension