Trees
Slater (1975) provides the following simple characterization of the metric dimension of a tree. If the tree is a path, its metric dimension is one. Otherwise, let L denote the set of degree-one vertices in the tree (usually called leaves, although Slater uses that word differently). Let K be the set of vertices that have degree greater than two, and that are connected by paths of degree-two vertices to one or more leaves. Then the metric dimension is |L| − |K|. A basis of this cardinality may be formed by removing from L one of the leaves associated with each vertex in K.
Read more about this topic: Metric Dimension (graph Theory)
Famous quotes containing the word trees:
“One wonders that the tithing-men and fathers of the town are not out to see what the trees mean by their high colors and exuberance of spirits, fearing that some mischief is brewing. I do not see what the Puritans did at this season, when the maples blaze out in scarlet. They certainly could not have worshiped in groves then. Perhaps that is what they built meeting-houses and fenced them round with horse-sheds for.”
—Henry David Thoreau (1817–1862)
“It was a tangled and perplexing thicket, through which we stumbled and threaded our way, and when we had finished a mile of it, our starting-point seemed far away. We were glad that we had not got to walk to Bangor along the banks of this river, which would be a journey of more than a hundred miles. Think of the denseness of the forest, the fallen trees and rocks, the windings of the river, the streams emptying in, and the frequent swamps to be crossed. It made you shudder.”
—Henry David Thoreau (1817–1862)