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A longest path between two given vertices s and t in a weighted graph G is the same thing as a shortest path in a graph −G derived from G by changing every weight to its negation. Therefore, if shortest paths can be found in −G, then longest paths can also be found in G. [4]
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).
In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. [1] A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. [2]
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.
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 ...
In an unweighted graph, the length of a cycle, path, or walk is the number of edges it uses. [4] In a weighted graph, it may instead be the sum of the weights of the edges that it uses. [5] Length is used to define the shortest path, girth (shortest cycle length), and longest path between two vertices in a graph.
A path is a particularly simple example of a tree, and in fact the paths are exactly the trees in which no vertex has degree 3 or more. A disjoint union of paths is called a linear forest . Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts.
Every two non-adjacent vertices can be connected by a rainbow path, either directly through the center vertex (bottom left) or by detouring around one triangle to avoid a repeated edge color (bottom right). In graph theory, a path in an edge-colored graph is said to be rainbow if no color repeats on it.
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