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On the metric dimension of Grassmann graphs
 Discrete Math. Theor. Comput. Sci
"... The metric dimension of a graph Γ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph Gq(n,k) (whose vertices are the ksubspaces of F n q, and are adjacent if they ..."
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The metric dimension of a graph Γ is the least number of vertices in a set with the property that the list of distances from any vertex to those in the set uniquely identifies that vertex. We consider the Grassmann graph Gq(n,k) (whose vertices are the ksubspaces of F n q, and are adjacent if they intersect in a (k − 1)subspace) for k ≥ 2. We find an upper bound on its metric dimension, which is equal to the number of 1dimensional subspaces of F n q. We also give a construction of a resolving set of this size in the case where k + 1 divides n, and a related construction in other cases. 1
On the complexity of metric dimension
 CORR, ABS/1107.2256. PROC. OF ESA 2012
"... The metric dimension of a graph G is the size of a smallest subset L ⊆ V (G) such that for any x, y ∈ V (G) there is a z ∈ L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, th ..."
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The metric dimension of a graph G is the size of a smallest subset L ⊆ V (G) such that for any x, y ∈ V (G) there is a z ∈ L such that the graph distance between x and z differs from the graph distance between y and z. Even though this notion has been part of the literature for almost 40 years, the computational complexity of determining the metric dimension of a graph is still very unclear. Essentially, we only know the problem to be NPhard for general graphs, to be polynomialtime solvable on trees, and to have a log napproximation algorithm for general graphs. In this paper, we show tight complexity boundaries for the Metric Dimension problem. We achieve this by giving two complementary results. First, we show that the Metric Dimension problem on boundeddegree planar graphs is NPcomplete. Then, we give a polynomialtime algorithm for determining the metric dimension of outerplanar graphs.
On the parameterized and approximation hardness of metric dimension
, 2012
"... The NPhard Metric Dimension problem is to decide for a given graph G and a positive integer k whether there is a vertex subset of size at most k that separates all vertex pairs in G. Herein, a vertex v separates a pair {u, w} if the distance (length of a shortest path) between v and u is different ..."
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The NPhard Metric Dimension problem is to decide for a given graph G and a positive integer k whether there is a vertex subset of size at most k that separates all vertex pairs in G. Herein, a vertex v separates a pair {u, w} if the distance (length of a shortest path) between v and u is different from the distance of v and w. We give a polynomialtime computable reduction from the Bipartite Dominating Set problem to Metric Dimension on maximum degree three graphs such that there is a onetoone correspondence between the solution sets of both problems. There are two main consequences of this: First, it proves that Metric Dimension on maximum degree three graphs is W[2]complete with respect to the parameter k. This answers an open question concerning the parameterized complexity of Metric Dimension posed by Díaz et al. [ESA’12] and already mentioned by Lokshtanov [Dagstuhl seminar, 2009]. Additionally, it implies that Metric Dimension cannot be solved in n o(k) time, unless the assumption FPT = W[1] fails. This proves that a trivial n O(k) algorithm is probably asymptotically optimal. Second, as Dominating Set is inapproximable within o(log n), it follows that Metric Dimension on maximum degree three graphs is also inapproximable by a factor of o(log n), unless NP = P. This strengthens the result of Hauptmann et al. [JDA’12] who proved APXhardness on boundeddegree graphs.
Distinguishability of maps
"... The distinguishing number of a group A acting faithfully on a set X, denoted D(A,X), is the least number of colors needed to color the elements of X so that no nonidentity element of A preserves the coloring. Given a map M (an embedding of a graph in a closed surface) with vertex set V and without l ..."
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The distinguishing number of a group A acting faithfully on a set X, denoted D(A,X), is the least number of colors needed to color the elements of X so that no nonidentity element of A preserves the coloring. Given a map M (an embedding of a graph in a closed surface) with vertex set V and without loops or multiples edges, let D(M) = D(Aut(M),V), where Aut(M) is the automorphism group of M; if M is orientable, define D + (M) similarly, using only orientationpreserving automorphisms. It is immediate that D(M) ≤ 4 and D + (M) ≤ 3. We use Russell and Sundaram’s Motion Lemma to show that there are only finitely many maps M with D(M)> 2. We show that if a group A of automorphisms of a graph G fixes no edges, then D(A,V) = 2, with five exceptions. That result is used to find the four maps with D + (M) = 3. We also consider the distinguishing chromatic number χD(M), where adjacent vertices get different colors. We show χD(M) ≤ χ(M) + 3 with equality in only finitely many cases, where χ(M) is the chromatic number of the graph underlying M. We also show that χD(M) ≤ 6 for planar maps, answering a question of Collins and Trenk. Finally, we discuss the implications for general group actions and give numerous problems for further study. 1
A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs
, 2014
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Resolving sets for Johnson and Kneser graphs
, 2012
"... 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) = 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 ..."
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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) = 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 to obtain resolving sets in these graphs, including (for Johnson graphs) projective planes and symmetric designs, as well as (for Kneser graphs) Hadamard matrices, Steiner systems, partial geometries and toroidal grids.
Metric dimension and zero forcing number of two families of line graphs
, 2012
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ON DISTANCES IN SIERPIŃSKI GRAPHS: ALMOSTEXTREME VERTICES AND METRIC DIMENSION
 APPL. ANAL. DISCRETE MATH. 7 (2013), 72–82
, 2013
"... Sierpiński graphs Snp form an extensively studied family of graphs of fractal nature applicable in topology, mathematics of the Tower of Hanoi, computer science, and elsewhere. An almostextreme vertex of Snp is introduced as a vertex that is either adjacent to an extreme vertex of Snp or is incide ..."
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Cited by 2 (1 self)
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Sierpiński graphs Snp form an extensively studied family of graphs of fractal nature applicable in topology, mathematics of the Tower of Hanoi, computer science, and elsewhere. An almostextreme vertex of Snp is introduced as a vertex that is either adjacent to an extreme vertex of Snp or is incident to an edge between two subgraphs of Snp isomorphic to S n−1 p. Explicit formulas are given for the distance in Snp between an arbitrary vertex and an almostextreme vertex. The formulas are applied to compute the total distance of almostextreme vertices and to obtain the metric dimension of Sierpiński graphs.