For an ordered set W = {w1, w2, . . . , wk} of vertices in a connected graph G and a vertex v of G, the code of v with respect to W is the k-vector cW (v) = (d(v, w1), d(v, w2), . . . , d(v, wk)). The setW is an independent resolving set for G if (1)W is independent in G and (2) distinct vertices have distinct codes with respect to W . The cardinality of a minimum independent resolving set in G...

In this writing, we point out some errors made in Boutin (Graphs Combin 25:789–806, 2009), where the author claims that a maximal independent set hereditary system is minimal determining (resolving) set. Further more, if exchange property holds at level of resolving sets, then, corresponding matroid. We give counter examples to disprove both her claims. Besides, prove there exist graphs having ...

A set of vertices S in a graph G is a resolving set for G if, for any two vertices u,v, there exists x ∈ S such that the distances d(u,x) 6= d(v,x). In this paper, we consider the Johnson graphs J(n,k) and Kneser graphs K(n,k), and obtain various constructions of resolving sets for these graphs. As well as general constructions, we show that various interesting combinatorial objects can be used...

The concept of (minimum) resolving set has proved to be useful and/or related to a variety of fields such as Chemistry [3,6], Robotic Navigation [5,8] and Combinatorial Search and Optimization [7]. This work is devoted to evaluating the so-called metric dimension of a finite connected graph, i.e., the minimum cardinality of a resolving set, for a number of graph families, as long as to study it...

For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W ) := (d(v, w1), d(v, w2), . . . , d(v, wk)) is called the metric representation of v with respect to W , where d(x, y) is the distance between vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W ...

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-locationdomination number. In this paper, we undertake a study that, in general graphs and specific families, relates metric-locating-dominating sets to other special sets: ...

A set of vertices S resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G H. We prove that the metric dimension of G G is tied in a strong sense to the minimum order of a so-called doubly resolving set ...

abstract bronchoalveolar carcinoma presenting as non-resolving consolidation is an uncommon presentation. the typical presentation of bronchoalveolar carcinoma is asymptomatic (solitary nodule) and remains without symptoms even as disease disseminates. we report a case of bronchoalveolar carcinoma presenting as non-resolving consolidation in a young male with productive cough, exertional breath...