6,518 research outputs found

    The neighbor-locating-chromatic number of pseudotrees

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    Ak-coloringof a graphGis a partition of the vertices ofGintokindependent sets,which are calledcolors. Ak-coloring isneighbor-locatingif any two vertices belongingto the same color can be distinguished from each other by the colors of their respectiveneighbors. Theneighbor-locating chromatic number¿NL(G) is the minimum cardinalityof a neighbor-locating coloring ofG.In this paper, we determine the neighbor-locating chromatic number of paths, cycles,fans and wheels. Moreover, a procedure to construct a neighbor-locating coloring ofminimum cardinality for these families of graphs is given. We also obtain tight upperbounds on the order of trees and unicyclic graphs in terms of the neighbor-locatingchromatic number. Further partial results for trees are also established.Preprin

    Characterizing All Trees with Locating-chromatic Number 3

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    Let cc be a proper kk-coloring of a connected graph GG. Let Π={S1,S2,,Sk}\Pi = \{S_{1}, S_{2},\ldots, S_{k}\} be the induced partition of V(G)V(G) by cc, where SiS_{i} is the partition class having all vertices with color ii.The color code cΠ(v)c_{\Pi}(v) of vertex vv is the orderedkk-tuple (d(v,S1),d(v,S2),,d(v,Sk))(d(v,S_{1}), d(v,S_{2}),\ldots, d(v,S_{k})), whered(v,Si)=min{d(v,x)xSi}d(v,S_{i})= \hbox{min}\{d(v,x)|x \in S_{i}\}, for 1ik1\leq i\leq k.If all vertices of GG have distinct color codes, then cc iscalled a locating-coloring of GG.The locating-chromatic number of GG, denoted by χL(G)\chi_{L}(G), isthe smallest kk such that GG posses a locating kk-coloring. Clearly, any graph of order n2n \geq 2 have locating-chromatic number kk, where 2kn2 \leq k \leq n. Characterizing all graphswith a certain locating-chromatic number is a difficult problem. Up to now, we have known allgraphs of order nn with locating chromatic number 2,n1,2, n-1, or nn.In this paper, we characterize all trees whose locating-chromatic number 33. We also give a family of trees with locating-chromatic number 4

    Locating-dominating sets in twin-free graphs

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    A locating-dominating set of a graph GG is a dominating set DD of GG with the additional property that every two distinct vertices outside DD have distinct neighbors in DD; that is, for distinct vertices uu and vv outside DD, N(u)DN(v)DN(u) \cap D \ne N(v) \cap D where N(u)N(u) denotes the open neighborhood of uu. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of GG, denoted γL(G)\gamma_L(G), is the minimum cardinality of a locating-dominating set in GG. 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 GG is a twin-free graph of order nn without isolated vertices, then γL(G)n2\gamma_L(G)\le \frac{n}{2}. We prove the general bound γL(G)2n3\gamma_L(G)\le \frac{2n}{3}, slightly improving over the 2n3+1\lfloor\frac{2n}{3}\rfloor+1 bound of Garijo et al. We then provide constructions of graphs reaching the n2\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 GG that are extremal for this bound. We finally prove the conjecture for split graphs and co-bipartite graphs.Comment: 11 pages; 4 figure

    Extensive facility location problems on networks with equity measures

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    AbstractThis paper deals with the problem of locating path-shaped facilities of unrestricted length on networks. We consider as objective functions measures conceptually related to the variability of the distribution of the distances from the demand points to a facility. We study the following problems: locating a path which minimizes the range, that is, the difference between the maximum and the minimum distance from the vertices of the network to a facility, and locating a path which minimizes a convex combination of the maximum and the minimum distance from the vertices of the network to a facility, also known in decision theory as the Hurwicz criterion. We show that these problems are NP-hard on general networks. For the discrete versions of these problems on trees, we provide a linear time algorithm for each objective function, and we show how our analysis can be extended also to the continuous case

    Trees with Certain Locating-chromatic Number

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    The locating-chromatic number of a graph G can be defined as the cardinality of a minimum resolving partition of the vertex set V(G) such that all vertices have distinct coordinates with respect to this partition and every two adjacent vertices in G are not contained in the same partition class. In this case, the coordinate of a vertex v in G is expressed in terms of the distances of v to all partition classes. This concept is a special case of the graph partition dimension notion. Previous authors have characterized all graphs of order n with locating-chromatic number either n or n-1. They also proved that there exists a tree of order n, n≥5, having locating-chromatic number k if and only if k âˆˆ{3,4,"¦,n-2,n}. In this paper, we characterize all trees of order n with locating-chromatic number n - t, for any integers n and t, where n > t+3 and 2 ≤ t < n/2

    Localization game on geometric and planar graphs

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    The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph GG we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant ζ(G)\zeta (G), defined as the least number of cops needed to localize the robber on a graph GG, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth 22 and unbounded ζ(G)\zeta (G). On a positive side, we prove that ζ(G)\zeta (G) is bounded by the pathwidth of GG. We then show that the algorithmic problem of determining ζ(G)\zeta (G) is NP-hard in graphs with diameter at most 22. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane

    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
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