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A polynomial-time approximation scheme is known for the minimum-diameter spanning tree in the plane. For any >, one can find a tree whose diameter is at most + times the optimum, in time (+). The algorithm involves approximating the input by the points of a coarse grid, chosen to give the best tree among a small number of grid orientations.
Deciding whether the metric dimension of a tree is at most a given integer can be done in linear time [10] Other linear-time algorithms exist for cographs, [5] chain graphs, [11] and cactus block graphs [12] (a class including both cactus graphs and block graphs). The problem may be solved in polynomial time on outerplanar graphs. [4]
The diameter of a graph can be computed by using a shortest path algorithm to compute shortest paths between all pairs of vertices, and then taking the maximum of the distances that it computes. For instance, in a graph with positive edge weights, this can be done by repeatedly using Dijkstra's algorithm , once for each possible starting vertex.
The Gabriel graph contains, as subgraphs, the Euclidean minimum spanning tree, the relative neighborhood graph, and the nearest neighbor graph. It is an instance of a beta-skeleton . Like beta-skeletons, and unlike Delaunay triangulations, it is not a geometric spanner : for some point sets, distances within the Gabriel graph can be much larger ...
The diameter d of a graph is the maximum eccentricity of any vertex in the graph. That is, d is the greatest distance between any pair of vertices or, alternatively, d = max v ∈ V ϵ ( v ) = max v ∈ V max u ∈ V d ( v , u ) . {\displaystyle d=\max _{v\in V}\epsilon (v)=\max _{v\in V}\max _{u\in V}d(v,u).}
When k is a fixed constant, the k-minimum spanning tree problem can be solved in polynomial time by a brute-force search algorithm that tries all k-tuples of vertices. However, for variable k, the k-minimum spanning tree problem has been shown to be NP-hard by a reduction from the Steiner tree problem. [1] [2]
Construct the shortest-path tree using the edges between each node and its parent. The above algorithm guarantees the existence of shortest-path trees. Like minimum spanning trees, shortest-path trees in general are not unique. In graphs for which all edge weights are equal, shortest path trees coincide with breadth-first search trees.
Two different tree-decompositions of the same graph. The width of a tree decomposition is the size of its largest set X i minus one. The treewidth tw(G) of a graph G is the minimum width among all possible tree decompositions of G. In this definition, the size of the largest set is diminished by one in order to make the treewidth of a tree ...