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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.
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 ...
A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges).
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]
Sum of the distance between the vertices and the difference of their colors is greater than k + 1, where k is a positive integer. Rank coloring If two vertices have the same color i, then every path between them contain a vertex with color greater than i Subcoloring An improper vertex coloring where every color class induces a union of cliques
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 ...
The latter may occur even if the distance in the other direction between the same two vertices is defined. In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path ...
there is a morphism between any two objects if and only if there is a (directed) path between the nodes, with the relation that this morphism is unique (any composition of maps is defined by its domain and target: this is the commutativity axiom). However, not every diagram commutes (the notion of diagram strictly generalizes commutative diagram).