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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.
In network science, a biased random walk on a graph is a time path process in which an evolving variable jumps from its current state to one of various potential new states; unlike in a pure random walk, the probabilities of the potential new states are unequal.
An elementary example of a random walk is the random walk on the integer number line which starts at 0, and at each step moves +1 or −1 with equal probability. Other examples include the path traced by a molecule as it travels in a liquid or a gas (see Brownian motion ), the search path of a foraging animal, or the price of a fluctuating ...
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).
Maximal entropy random walk (MERW) is a popular type of biased random walk on a graph, in which transition probabilities are chosen accordingly to the principle of maximum entropy, which says that the probability distribution which best represents the current state of knowledge is the one with largest entropy.
An Eulerian trail, [note 1] or Euler walk, in an undirected graph is a walk that uses each edge exactly once. If such a walk exists, the graph is called traversable or semi-eulerian. [3] An Eulerian cycle, [note 1] also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once.
In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G. Thus, finding a Hamiltonian path cannot be significantly slower (in the worst case, as a function of the number of vertices) than finding a ...
A Hamiltonian cycle around a network of six vertices Examples of Hamiltonian cycles on a square grid graph 8x8. In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once.