116 research outputs found
Metric Dimension of a Diagonal Family of Generalized Hamming Graphs
Classical Hamming graphs are Cartesian products of complete graphs, and two
vertices are adjacent if they differ in exactly one coordinate. Motivated by
connections to unitary Cayley graphs, we consider a generalization where two
vertices are adjacent if they have no coordinate in common. Metric dimension of
classical Hamming graphs is known asymptotically, but, even in the case of
hypercubes, few exact values have been found. In contrast, we determine the
metric dimension for the entire diagonal family of -dimensional generalized
Hamming graphs. Our approach is constructive and made possible by first
characterizing resolving sets in terms of forbidden subgraphs of an auxiliary
edge-colored hypergraph.Comment: 19 pages, 7 figure
Further Contributions on the Outer Multiset Dimension of Graphs
The outer multiset dimension dim ms(G) of a graph G is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that dim ms(G) = n(G) - 1 if and only if G is a regular graph with diameter at most 2. Graphs G with dim ms(G) = 2 are described and recognized in polynomial time. A lower bound on the lexicographic product of G and H is proved when H is complete or edgeless, and the extremal graphs are determined. It is proved that dimms(Ps□Pt)=3 for s≥ t≥ 2.15 página
Centroidal bases in graphs
We introduce the notion of a centroidal locating set of a graph , that is,
a set of vertices such that all vertices in are uniquely determined by
their relative distances to the vertices of . A centroidal locating set of
of minimum size is called a centroidal basis, and its size is the
centroidal dimension . 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 is lower- and upper-bounded by the metric
dimension and twice the location-domination number of , respectively. The
latter two parameters are standard and well-studied notions in the field of
graph identification.
We show that for any graph with vertices and maximum degree at
least~2, . 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,
, the bound being tight for paths and
cycles. We finally investigate the computational complexity of determining
for an input graph , showing that the problem is hard and cannot
even be approximated efficiently up to a factor of . We also give an
-approximation algorithm
On the robustness of the metric dimension of grid graphs to adding a single edge
The metric dimension (MD) of a graph is a combinatorial notion capturing the
minimum number of landmark nodes needed to distinguish every pair of nodes in
the graph based on graph distance. We study how much the MD can increase if we
add a single edge to the graph. The extra edge can either be selected
adversarially, in which case we are interested in the largest possible value
that the MD can take, or uniformly at random, in which case we are interested
in the distribution of the MD. The adversarial setting has already been studied
by [Eroh et. al., 2015] for general graphs, who found an example where the MD
doubles on adding a single edge. By constructing a different example, we show
that this increase can be as large as exponential. However, we believe that
such a large increase can occur only in specially constructed graphs, and that
in most interesting graph families, the MD at most doubles on adding a single
edge. We prove this for -dimensional grid graphs, by showing that
appropriately chosen corners and the endpoints of the extra edge can
distinguish every pair of nodes, no matter where the edge is added. For the
special case of , we show that it suffices to choose the four corners as
landmarks. Finally, when the extra edge is sampled uniformly at random, we
conjecture that the MD of 2-dimensional grids converges in probability to
, and we give an almost complete proof
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Diamond-based models for scientific visualization
Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains including scientific visualization, finite element analysis and shape modeling and analysis. A popular class of such approaches is based on the regular simplex bisection operator, which bisects simplices (e.g. line segments, triangles, tetrahedra) along the midpoint of a predetermined edge. Regular simplex bisection produces adaptive simplicial meshes of high geometric quality, while simplifying the extraction of crack-free, or conforming, approximations to the original dataset. Efficient multiresolution representations for such models have been achieved in 2D and 3D by clustering sets of simplices sharing the same bisection edge into structures called diamonds. In this thesis, we introduce several diamond-based approaches for scientific visualization. We first formalize the notion of diamonds in arbitrary dimensions in terms of two related simplicial decompositions of hypercubes. This enables us to enumerate the vertices, simplices, parents and children of a diamond. In particular, we identify the number of simplices involved in conforming updates to be factorial in the dimension and group these into a linear number of subclusters of simplices that are generated simultaneously. The latter form the basis for a compact pointerless representation for conforming meshes generated by regular simplex bisection and for efficiently navigating the topological connectivity of these meshes. Secondly, we introduce the supercube as a high-level primitive on such nested meshes based on the atomic units within the underlying triangulation grid. We propose the use of supercubes to associate information with coherent subsets of the full hierarchy and demonstrate the effectiveness of such a representation for modeling multiresolution terrain and volumetric datasets. Next, we introduce Isodiamond Hierarchies, a general framework for spatial access structures on a hierarchy of diamonds that exploits the implicit hierarchical and geometric relationships of the diamond model. We use an isodiamond hierarchy to encode irregular updates to a multiresolution isosurface or interval volume in terms of regular updates to diamonds. Finally, we consider nested hypercubic meshes, such as quadtrees, octrees and their higher dimensional analogues, through the lens of diamond hierarchies. This allows us to determine the relationships involved in generating balanced hypercubic meshes and to propose a compact pointerless representation of such meshes. We also provide a local diamond-based triangulation algorithm to generate high-quality conforming simplicial meshes
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