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 distanc es from the elements of S , and the minimum cardinality of such a set is called the metri c-location- domination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominatin g sets to other special sets: resolving sets, dominating sets, locating-dominating set s and doubly resolving sets. We first characterize classes of trees according to cer tain relationships between their metric-location-domination number and thei r metric dimension and domination number. Then, we show different methods to tran sform metric- locating-dominating sets into locating-dominating sets a nd 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 farPostprint (published version

Metric-locating-dominating sets of graphs for constructing related subsets of vertices

© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/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 the extremal trees of the bounds that naturally involve metric-location-domination number, metric dimension and domination number. Then, we prove that there is no polynomial upper bound on the location-domination number in terms of the metric-location-domination number, thus extending a result of Henning and Oellermann. Finally, 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 concerning parameters that have not been related so farPeer ReviewedPostprint (author's final draft

The equidistant dimension of graphs

A subset S of vertices of a connected graph G is a distance-equalizer set if for every two distinct vertices x,y¿V(G)\S there is a vertex w¿S such that the distances from x and y to w are the same. The equidistant dimension of G is the minimum cardinality of a distance-equalizer set of G. This paper is devoted to introduce this parameter and explore its properties and applications to other mathematical problems, not necessarily in the context of graph theory. Concretely, we first establish some bounds concerning the order, the maximum degree, the clique number, and the independence number, and characterize all graphs attaining some extremal values. We then study the equidistant dimension of several families of graphs (complete and complete multipartite graphs, bistars, paths, cycles, and Johnson graphs), proving that, in the case of paths and cycles, this parameter is related to 3-AP-free sets. Subsequently, we show the usefulness of distance-equalizer sets for constructing doubly resolving sets.Peer ReviewedPostprint (published version

Dominating 2-broadcast in graphs: complexity, bounds and extremal graphs

Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating 2-broadcasts along with the associated parameter, the dominating 2-broadcast number. We prove that computing the dominating 2-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desired.Peer ReviewedPostprint (author's final draft

On global location-domination in graphs

A dominating set S of a graph G is called locating-dominating, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number lambda(G). An LD-set S of a graph G is global if it is an LD-set of both G and its complement G'. The global location-domination number lambda g(G) is introduced as the minimum cardinality of a global LD-set of G. In this paper, some general relations between LD-codes and the location-domination number in a graph and its complement are presented first. Next, a number of basic properties involving the global location-domination number are showed. Finally, both parameters are studied in-depth for the family of block-cactus graphs.Postprint (published version

Metric-locating-dominating partitions in graphs

A partition ¿ = { S 1 ,...,S k } of the vertex set of a connected graph G is a metric-locating partition of G if for every pair of vertices u,v belonging to the same part S i , d ( u,S j ) 6 = d ( v,S j ), for some other part S j . The partition dimension ß p ( G ) is the minimum cardinality of a metric- locating partition of G . A metric-locating partition ¿ is called metric-locating-dominanting if for every vertex v of G , d ( v,S j ) = 1, for some part S j of ¿. The partition metric-location-domination number ¿ p ( G ) is the minimum cardinality of a metric-locating-dominating partition of G . In this paper we show, among other results, that ß p ( G ) = ¿ p ( G ) = ß p ( G ) + 1. We also charac- terize all connected graphs of order n = 7 satisfying any of the following conditions: ¿ p ( G ) = n - 1, ¿ p ( G ) = n - 2 and ß p ( G ) = n - 2. Finally, we present some tight Nordhaus-Gaddum bounds for both the partition dimension ß ( G ) and the partition metric-location-domination number ¿ ( G ). Keywords: dominating partition, locating partition, location, domination, metric locationPeer ReviewedPostprint (author's final draft

Neighbor-locating coloring: graph operations and extremal cardinalities

© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/A k–coloring of a graph is a k-partition of V into independent sets, called colors. A k-coloring is called neighbor-locating if for every pair of vertices u, v belonging to the same color , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v. The neighbor-locating chromatic number, , is the minimum cardinality of a neighbor-locating coloring of G. In this paper, we examine the neighbor-locating chromatic number for various graph operations: the join, the disjoint union and Cartesian product. We also characterize all connected graphs of order with neighbor-locating chromatic number equal either to n or to and determine the neighbor-locating chromatic number of split graphs.Peer ReviewedPostprint (author's final draft

On perfect and quasiperfect dominations in graphs

A subset S ¿ V in a graph G = ( V , E ) is a k -quasiperfect dominating set (for k = 1) if every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k -quasiperfect dominating set in G is denoted by ¿ 1 k ( G ). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n = ¿ 11 ( G ) = ¿ 12 ( G ) = ... = ¿ 1 ¿ ( G ) = ¿ ( G ) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ¿ 12 ( G ) = ¿ ( G ). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ¿ ( G ).Postprint (published version

Graphs of non-crossing perfect matchings

Let Pn be a set of n = 2m points that are the vertices of a convex polygon, and let Mm be the graph having as vertices all the perfect matchings in the point set Pn whose edges are straight line segments and do not cross, and edges joining two perfect matchings M1 and M2 if M2 = M1 ¡ (a; b) ¡ (c; d) + (a; d) + (b; c) for some points a; b; c; d of Pn. We prove the following results about Mm: its diameter is m ¡ 1; it is bipartite for every m; the connectivity is equal to m ¡ 1; it has no Hamilton path for m odd, m > 3; and finally it has a Hamilton cycle for every m even, m>=4

Locating domination in graphs and their complements

A dominating set S of a graph G is called locating-dominating , LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S . Locating-dominating sets of minimum cardinality are called LD -codes and the cardinality of an LD-code is the location-domination number . An LD-set of a graph G is global if S is an LD-set of both G and its complement, G . In this work, we give some relations between the locating-dominating sets and location-domination number in a graph and its complementPostprint (published version