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In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges.
Fig 1. Construction of the first isogonic center, X(13). When no angle of the triangle exceeds 120°, this point is the Fermat point. In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest possible [1] or ...
Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph. A related problem is to find a partition that is optimal terms of the number of edges between parts ...
The above theorem can only recognize the existence of a Hamiltonian path in a graph and not a Hamiltonian Cycle. Many of these results have analogues for balanced bipartite graphs, in which the vertex degrees are compared to the number of vertices on a single side of the bipartition rather than the number of vertices in the whole graph. [13]
K n can be decomposed into n trees T i such that T i has i vertices. [6] Ringel's conjecture asks if the complete graph K 2n+1 can be decomposed into copies of any tree with n edges. [7] This is known to be true for sufficiently large n. [8] [9] The number of all distinct paths between a specific pair of vertices in K n+2 is given [10] by
The general graph Steiner tree problem can be approximated by computing the minimum spanning tree of the subgraph of the metric closure of the graph induced by the terminal vertices, as first published in 1981 by Kou et al. [18] The metric closure of a graph G is the complete graph in which each edge is weighted by the shortest path distance ...
Conversely, if H has an induced path or cycle of length k, any maximal set of nonadjacent vertices in G from this path or cycle forms an independent set in G of size at least k/3. Thus, the size of the maximum independent set in G is within a constant factor of the size of the longest induced path and the longest induced cycle in H.
Van Schooten's theorem, named after the Dutch mathematician Frans van Schooten, describes a property of equilateral triangles.It states: For an equilateral triangle with a point on its circumcircle the length of longest of the three line segments ,, connecting with the vertices of the triangle equals the sum of the lengths of the other two.