41,941 research outputs found
Metric Dimension: from Graphs to Oriented Graphs
International audienceThe metric dimension of an undirected graph is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of . Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs. In this work, we study, for particular graph families, the maximum metric dimension over all strongly-connected orientations, by exhibiting lower and upper bounds on this value. We first exhibit general bounds for graphs with bounded maximum degree. In particular, we prove that, in the case of subcubic -node graphs, all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension . We then consider strongly-connected orientations of grids. For a torus with rows and columns, we show that the maximum value of the metric dimension of a strongly-connected Eulerian orientation is asymptotically (the equality holding when , are even, which is best possible). For a grid with rows and columns, we prove that all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension
Metric Dimension: from Graphs to Oriented Graphs
International audienceThe metric dimension of an undirected graph is the cardinality of a smallest set of vertices that allows, through their distances to all vertices, to distinguish any two vertices of . Many aspects of this notion have been investigated since its introduction in the 70's, including its generalization to digraphs. In this work, we study, for particular graph families, the maximum metric dimension over all strongly-connected orientations, by exhibiting lower and upper bounds on this value. We first exhibit general bounds for graphs with bounded maximum degree. In particular, we prove that, in the case of subcubic -node graphs, all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension . We then consider strongly-connected orientations of grids. For a torus with rows and columns, we show that the maximum value of the metric dimension of a strongly-connected Eulerian orientation is asymptotically (the equality holding when , are even, which is best possible). For a grid with rows and columns, we prove that all strongly-connected orientations asymptotically have metric dimension at most , and that there are such orientations having metric dimension
Connectivity of orientations of 3-edge-connected graphs
We attempt to generalize a theorem of Nash-Williams stating that a graph has
a -arc-connected orientation if and only if it is -edge-connected. In a
strongly connected digraph we call an arc {\it deletable} if its deletion
leaves a strongly connected digraph. Given a -edge-connected graph , we
define its Frank number to be the minimum number such that there
exist orientations of with the property that every edge becomes a
deletable arc in at least one of these orientations. We are interested in
finding a good upper bound for the Frank number. We prove that for
every -edge-connected graph. On the other hand, we show that a Frank number
of is attained by the Petersen graph. Further, we prove better upper bounds
for more restricted classes of graphs and establish a connection to the
Berge-Fulkerson conjecture. We also show that deciding whether all edges of a
given subset can become deletable in one orientation is NP-complete.Comment: to appear in European Journal of Combinatoric
Graphs with many strong orientations
We establish mild conditions under which a possibly irregular, sparse graph
has "many" strong orientations. Given a graph on vertices, orient
each edge in either direction with probability independently. We show
that if satisfies a minimum degree condition of and has
Cheeger constant at least , then the
resulting randomly oriented directed graph is strongly connected with high
probability. This Cheeger constant bound can be replaced by an analogous
spectral condition via the Cheeger inequality. Additionally, we provide an
explicit construction to show our minimum degree condition is tight while the
Cheeger constant bound is tight up to a factor.Comment: 14 pages, 4 figures; revised version includes more background and
minor changes that better clarify the expositio
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