On Domination Number and Distance in Graphs

A vertex set $S$ of a graph $G$ is a \emph{dominating set} if each vertex of $G$ either belongs to $S$ or is adjacent to a vertex in $S$. The \emph{domination number} $\gamma(G)$ of $G$ is the minimum cardinality of $S$ as $S$ varies over all dominating sets of $G$. It is known that $\gamma(G) \ge \frac{1}{3}(diam(G)+1)$, where $diam(G)$ denotes the diameter of $G$. Define $C_r$ as the largest constant such that $\gamma(G) \ge C_r \sum_{1 \le i < j \le r}d(x_i, x_j)$ for any $r$ vertices of an arbitrary connected graph $G$; then $C_2=\frac{1}{3}$ in this view. The main result of this paper is that $C_r=\frac{1}{r(r-1)}$ for $r\geq 3$. It immediately follows that $\gamma(G)\geq \mu(G)=\frac{1}{n(n-1)}W(G)$, where $\mu(G)$ and $W(G)$ are respectively the average distance and the Wiener index of $G$ of order $n$. As an application of our main result, we prove a conjecture of DeLaVi\~{n}a et al.\;that $\gamma(G)\geq \frac{1}{2}(ecc_G(B)+1)$, where $ecc_G(B)$ denotes the eccentricity of the boundary of an arbitrary connected graph $G$.Comment: 5 pages, 2 figure

Resolving sets for breaking symmetries of graphs

This paper deals with the maximum value of the difference between the determining number and the metric dimension of a graph as a function of its order. Our technique requires to use locating-dominating sets, and perform an independent study on other functions related to these sets. Thus, we obtain lower and upper bounds on all these functions by means of very diverse tools. Among them are some adequate constructions of graphs, a variant of a classical result in graph domination and a polynomial time algorithm that produces both distinguishing sets and determining sets. Further, we consider specific families of graphs where the restrictions of these functions can be computed. To this end, we utilize two well-known objects in graph theory: $k$-dominating sets and matchings.Comment: 24 pages, 12 figure

Locating-dominating sets: From graphs to oriented graphs

A locating-dominating set of an undirected graph is a subset of vertices S such that S is dominating and for every u, v is not an element of S, the neighbourhood of u and v on S are distinct (i.e. N(u) & cap; S &NOTEQUexpressionL; N(v) & cap; S). Locating-dominating sets have received a considerable attention in the last decades. In this paper, we consider the oriented version of the problem. A locating-dominating set in an oriented graph is a set S such that for each w is an element of V \ S, N-(w) & cap; S &NOTEQUexpressionL; Phi and for each pair of distinct vertices u, v is an element of V \ S, N-(u) & cap; S &NOTEQUexpressionL; N-(v) & cap; S. We consider the following two parameters. Given an undirected graph G, we look for (gamma)over the arrow(LD) (G) ((gamma)over the arrow(LD) (G)) which is the size of the smallest (largest) optimal locating-dominating set over all orientations of G. In particular, if D is an orientation of G, then (gamma)over the arrow(LD)(G) = gamma(LD)(G) and some for which (gamma)over the arrow(LD)(G) = alpha(G). Finally, we show that for many graph classes (gamma)over the arrow(LD)(G) is polynomial on n but we leave open the question whether there exist graphs with (gamma)over the arrow(LD)(G) is an element of O (log n). (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).</p