889 research outputs found

    The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families

    Get PDF
    We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as V(G), and the following terminology. Two vertices u,v is an element of V(G) are strongly resolved by a vertex w is an element of V(G), if there is a shortest w-v path containing u or a shortest w-u containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family F of graphs defined over a common vertex set V, a set S subset of V is an SSMG for F, if such set S is a strong metric generator for every graph G is an element of F. The simultaneous strong metric dimension of F is the minimum cardinality of any strong metric generator for F, and is denoted by Sds(F). The notion of simultaneous strong resolving graph of a graph family F is introduced in this work, and its usefulness in the study of Sds(F) is described. That is, it is proved that computing Sds(F) is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of F. Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature

    On the strong partition dimension of graphs

    Full text link
    We present a different way to obtain generators of metric spaces having the property that the ``position'' of every element of the space is uniquely determined by the distances from the elements of the generators. Specifically we introduce a generator based on a partition of the metric space into sets of elements. The sets of the partition will work as the new elements which will uniquely determine the position of each single element of the space. A set WW of vertices of a connected graph GG strongly resolves two different vertices x,yWx,y\notin W if either dG(x,W)=dG(x,y)+dG(y,W)d_G(x,W)=d_G(x,y)+d_G(y,W) or dG(y,W)=dG(y,x)+dG(x,W)d_G(y,W)=d_G(y,x)+d_G(x,W), where dG(x,W)=min{d(x,w)  :  wW}d_G(x,W)=\min\left\{d(x,w)\;:\;w\in W\right\}. An ordered vertex partition Π={U1,U2,...,Uk}\Pi=\left\{U_1,U_2,...,U_k\right\} of a graph GG is a strong resolving partition for GG if every two different vertices of GG belonging to the same set of the partition are strongly resolved by some set of Π\Pi. A strong resolving partition of minimum cardinality is called a strong partition basis and its cardinality the strong partition dimension. In this article we introduce the concepts of strong resolving partition and strong partition dimension and we begin with the study of its mathematical properties. We give some realizability results for this parameter and we also obtain tight bounds and closed formulae for the strong metric dimension of several graphs.Comment: 16 page

    Defensive alliances in graphs: a survey

    Full text link
    A set SS of vertices of a graph GG is a defensive kk-alliance in GG if every vertex of SS has at least kk more neighbors inside of SS than outside. This is primarily an expository article surveying the principal known results on defensive alliances in graph. Its seven sections are: Introduction, Computational complexity and realizability, Defensive kk-alliance number, Boundary defensive kk-alliances, Defensive alliances in Cartesian product graphs, Partitioning a graph into defensive kk-alliances, and Defensive kk-alliance free sets.Comment: 25 page

    Open k-monopolies in graphs: complexity and related concepts

    Get PDF
    Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open kk-monopolies in graphs which are closely related to different parameters in graphs. Given a graph G=(V,E)G=(V,E) and XVX\subseteq V, if δX(v)\delta_X(v) is the number of neighbors vv has in XX, kk is an integer and tt is a positive integer, then we establish in this article a connection between the following three concepts: - Given a nonempty set MVM\subseteq V a vertex vv of GG is said to be kk-controlled by MM if δM(v)δV(v)2+k\delta_M(v)\ge \frac{\delta_V(v)}{2}+k. The set MM is called an open kk-monopoly for GG if it kk-controls every vertex vv of GG. - A function f:V{1,1}f: V\rightarrow \{-1,1\} is called a signed total tt-dominating function for GG if f(N(v))=vN(v)f(v)tf(N(v))=\sum_{v\in N(v)}f(v)\geq t for all vVv\in V. - A nonempty set SVS\subseteq V is a global (defensive and offensive) kk-alliance in GG if δS(v)δVS(v)+k\delta_S(v)\ge \delta_{V-S}(v)+k holds for every vVv\in V. In this article we prove that the problem of computing the minimum cardinality of an open 00-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs. In addition we present some general bounds for the minimum cardinality of open kk-monopolies and we derive some exact values.Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016
    corecore