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The median of three vertices in a tree, showing the subtree formed by the union of shortest paths between the vertices. Every tree is a median graph. To see this, observe that in a tree, the union of the three shortest paths between pairs of the three vertices a, b, and c is either itself a path, or a subtree formed by three paths meeting at a single central node with degree three.
The squaregraphs include as special cases trees, grid graphs, gear graphs, and the graphs of polyominos.. As well as being planar graphs, squaregraphs are median graphs, meaning that for every three vertices u, v, and w there is a unique median vertex m(u,v,w) that lies on shortest paths between each pair of the three vertices. [1]
Download as PDF; Printable version; In other projects ... Glossary of graph theory terms; Undirected graph; ... Unit disk graph; Line graph; Claw-free graph; Median ...
The trees and hypercube graphs are examples of median graphs. Since the median graphs include the squaregraphs, simplex graphs, and Fibonacci cubes, as well as the covering graphs of finite distributive lattices, these are all partial cubes. The planar dual graph of an arrangement of lines in the Euclidean plane is a partial cube.
A median graph is an undirected graph in which for every three vertices , , and there is a unique vertex ,, that belongs to shortest paths between any two of , , and . If this is the case, then the operation x , y , z {\displaystyle \langle x,y,z\rangle } defines a median algebra having the vertices of the graph as its elements.
For the 1-dimensional case, the geometric median coincides with the median.This is because the univariate median also minimizes the sum of distances from the points. (More precisely, if the points are p 1, ..., p n, in that order, the geometric median is the middle point (+) / if n is odd, but is not uniquely determined if n is even, when it can be any point in the line segment between the two ...
Reinhard Diestel (born 1959) [1] is a German mathematician specializing in graph theory, including the interplay among graph minors, matroid theory, tree decomposition, and infinite graphs. He holds the chair of discrete mathematics at the University of Hamburg .
Median (geometry), in geometry, a line joining a vertex of a triangle to the midpoint of the opposite side; Median (graph theory), a vertex m(a,b,c) that belongs to shortest paths between each pair of a, b, and c; Median algebra, an algebraic triple product generalising the algebraic properties of the majority function