7,127 research outputs found

    Centroidal bases in graphs

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    We introduce the notion of a centroidal locating set of a graph GG, that is, a set LL of vertices such that all vertices in GG are uniquely determined by their relative distances to the vertices of LL. A centroidal locating set of GG of minimum size is called a centroidal basis, and its size is the centroidal dimension CD(G)CD(G). This notion, which is related to previous concepts, gives a new way of identifying the vertices of a graph. The centroidal dimension of a graph GG is lower- and upper-bounded by the metric dimension and twice the location-domination number of GG, respectively. The latter two parameters are standard and well-studied notions in the field of graph identification. We show that for any graph GG with nn vertices and maximum degree at least~2, (1+o(1))lnnlnlnnCD(G)n1(1+o(1))\frac{\ln n}{\ln\ln n}\leq CD(G) \leq n-1. We discuss the tightness of these bounds and in particular, we characterize the set of graphs reaching the upper bound. We then show that for graphs in which every pair of vertices is connected via a bounded number of paths, CD(G)=Ω(E(G))CD(G)=\Omega\left(\sqrt{|E(G)|}\right), the bound being tight for paths and cycles. We finally investigate the computational complexity of determining CD(G)CD(G) for an input graph GG, showing that the problem is hard and cannot even be approximated efficiently up to a factor of o(logn)o(\log n). We also give an O(nlnn)O\left(\sqrt{n\ln n}\right)-approximation algorithm

    Locating-dominating sets and identifying codes in graphs of girth at least 5

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    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

    Resolving sets for breaking symmetries of graphs

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    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: kk-dominating sets and matchings.Comment: 24 pages, 12 figure

    New results on metric-locating-dominating sets of graphs

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    A dominating set SS 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 SS, 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

    New results on metric-locating-dominating sets of graphs

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    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

    Coloring, location and domination of corona graphs

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    A vertex coloring of a graph GG is an assignment of colors to the vertices of GG such that every two adjacent vertices of GG 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 SS of vertices of a graph GG is a dominating set in GG if every vertex outside of SS is adjacent to at least one vertex belonging to SS. A domination parameter of GG is related to those structures of a graph satisfying some domination property together with other conditions on the vertices of GG. In this article we study several mathematical properties related to coloring, domination and location of corona graphs. We investigate the distance-kk 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-kk chromatic number of corona graphs and the distance-kk chromatic number of its factors. Moreover, we give the exact value of the distance-kk 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 kk-domination number, the distance-kk domination number, the independence domination number, the domatic number and the idomatic number of corona graphs.Comment: 18 page
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