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An induced path of length four in a cube. Finding the longest induced path in a hypercube is known as the snake-in-the-box problem. In the mathematical area of graph theory, an induced path in an undirected graph G is a path that is an induced subgraph of G.
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 graph theory terminology, this is called finding the longest possible induced path in a hypercube; it can be viewed as a special case of the induced subgraph isomorphism problem. There is a similar problem of finding long induced cycles in hypercubes, called the coil-in-the-box problem.
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).
A path in which all the edge s have the same direction. If a directed path leads from vertex x to vertex y, x is a predecessor of y, y is a successor of x, and y is said to be reachable from x. direction 1. The asymmetric relation between two adjacent vertices in a graph, represented as an arrow. 2.
Snake-in-the-box, the longest induced path in a hypercube; Steinhaus–Johnson–Trotter algorithm for finding a Hamiltonian path in a permutohedron; Subhamiltonian graph, a subgraph of a planar Hamiltonian graph; Tait's conjecture (now known false) that 3-regular polyhedral graphs are Hamiltonian; Travelling salesman problem
The graphs G i may be taken as the induced subgraphs of the sets X i in the first definition of path decompositions, with two vertices in successive induced subgraphs being glued together when they are induced by the same vertex in G, and in the other direction one may recover the sets X i as the vertex sets of the graphs G i.
The shortest path between any two vertices in an unweighted graph is always an induced path, because any additional edges between pairs of vertices that could cause it to be not induced would also cause it to be not shortest. Conversely, in distance-hereditary graphs, every induced path is a shortest path. [2]