Locating and Total Dominating Sets of Direct Products of Complete Graphs

A set S of vertices in a graph G = (V,E) is a metric-locating-total dominating set of G if every vertex of V is adjacent to a vertex in S and for every u ≠ v in V there is a vertex x in S such that d(u,x) ≠ d(v,x). The metric-location-total domination number \gamma^M_t(G) of G is the minimum cardinality of a metric-locating-total dominating set in G. For graphs G and H, the direct product G × H is the graph with vertex set V(G) × V(H) where two vertices (x,y) and (v,w) are adjacent if and only if xv in E(G) and yw in E(H). In this paper, we determine the lower bound of the metric-location-total domination number of the direct products of complete graphs. We also determine some exact values for some direct products of two complete graphs

Locating-total dominating sets in twin-free graphs: a conjecture

A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$. A locating-total dominating set of $G$ is a total dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-total domination number of $G$, denoted $LT(G)$, is the minimum cardinality of a locating-total dominating set in $G$. It is well-known that every connected graph of order $n \geq 3$ has a total dominating set of size at most $\frac{2}{3}n$. We conjecture that if $G$ is a twin-free graph of order $n$ with no isolated vertex, then $LT(G) \leq \frac{2}{3}n$. We prove the conjecture for graphs without $4$-cycles as a subgraph. We also prove that if $G$ is a twin-free graph of order $n$, then $LT(G) \le \frac{3}{4}n$.Comment: 18 pages, 1 figur

Location-domination in line graphs

A set $D$ of vertices of a graph $G$ is locating if every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \neq N(v) \cap D$, where $N(u)$ denotes the open neighborhood of $u$. If $D$ is also a dominating set (total dominating set), it is called a locating-dominating set (respectively, locating-total dominating set) of $G$. A graph $G$ is twin-free if every two distinct vertices of $G$ have distinct open and closed neighborhoods. It is conjectured [D. Garijo, A. Gonzalez and A. Marquez, The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] and [F. Foucaud and M. A. Henning. Locating-total dominating sets in twin-free graphs: a conjecture. The Electronic Journal of Combinatorics 23 (2016), P3.9] respectively, that any twin-free graph $G$ without isolated vertices has a locating-dominating set of size at most one-half its order and a locating-total dominating set of size at most two-thirds its order. In this paper, we prove these two conjectures for the class of line graphs. Both bounds are tight for this class, in the sense that there are infinitely many connected line graphs for which equality holds in the bounds.Comment: 23 pages, 2 figure

Locating-dominating sets in twin-free graphs

A locating-dominating set of a graph $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \ne N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of $G$, denoted $\gamma_L(G)$, is the minimum cardinality of a locating-dominating set in $G$. It is conjectured [D. Garijo, A. Gonz\'alez and A. M\'arquez. The difference between the metric dimension and the determining number of a graph. Applied Mathematics and Computation 249 (2014), 487--501] that if $G$ is a twin-free graph of order $n$ without isolated vertices, then $\gamma_L(G)\le \frac{n}{2}$. We prove the general bound $\gamma_L(G)\le \frac{2n}{3}$, slightly improving over the $\lfloor\frac{2n}{3}\rfloor+1$ bound of Garijo et al. We then provide constructions of graphs reaching the $\frac{n}{2}$ bound, showing that if the conjecture is true, the family of extremal graphs is a very rich one. Moreover, we characterize the trees $G$ that are extremal for this bound. We finally prove the conjecture for split graphs and co-bipartite graphs.Comment: 11 pages; 4 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 and identifying codes in graphs of girth at least 5

Locating-dominating sets and identifying codes are two closely related notions in the area of separating systems. Roughly speaking, they consist in a dominating set of a graph such that every vertex is uniquely identified by its neighbourhood within the dominating set. In this paper, we study the size of a smallest locating-dominating set or identifying code for graphs of girth at least 5 and of given minimum degree. We use the technique of vertex-disjoint paths to provide upper bounds on the minimum size of such sets, and construct graphs who come close to meet these bounds.Comment: 20 pages, 9 figure

Coloring, location and domination of corona graphs

A vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that every two adjacent vertices of $G$ have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the vertices of a graph, in which the process of labeling is done according to an extra condition. A set $S$ of vertices of a graph $G$ is a dominating set in $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. A domination parameter of $G$ is related to those structures of a graph satisfying some domination property together with other conditions on the vertices of $G$. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-$k$ colorings of corona graphs. Particularly, we obtain tight bounds for the distance-2 chromatic number and distance-3 chromatic number of corona graphs, throughout some relationships between the distance-$k$ chromatic number of corona graphs and the distance-$k$ chromatic number of its factors. Moreover, we give the exact value of the distance-$k$ chromatic number of the corona of a path and an arbitrary graph. On the other hand, we obtain bounds for the Roman dominating number and the locating-domination number of corona graphs. We give closed formulaes for the $k$-domination number, the distance-$k$ domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.Comment: 18 page

New results on metric-locating-dominating sets of graphs

A dominating set $S$ of a graph is a metric-locating-dominating set if each vertex of the graph is uniquely distinguished by its distances from the elements of $S$, and the minimum cardinality of such a set is called the metric-location-domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: resolving sets, dominating sets, locating-dominating sets and doubly resolving sets. We first characterize classes of trees according to certain relationships between their metric-location-domination number and their metric dimension and domination number. Then, we show different methods to transform metric-locating-dominating sets into locating-dominating sets and doubly resolving sets. Our methods produce new bounds on the minimum cardinalities of all those sets, some of them involving parameters that have not been related so far.Comment: 13 pages, 3 figure