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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]
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.
Two internally disjoint paths are edge-disjoint, but the converse is not necessarily true. The distance between two vertices in a graph is the length of a shortest path between them, if one exists, and otherwise the distance is infinity. The diameter of a connected graph is the largest distance (defined above) between pairs of vertices of the ...
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 ...
A bipartite graph may be oriented from one side of the bipartition to the other. The longest path in this orientation has length one, with only two vertices. Conversely, if a graph is oriented without any three-vertex paths, then every vertex must either be a source (with no incoming edges) or a sink (with no outgoing edges) and the partition of the vertices into sources and sinks shows that ...
The center is the middle vertex or middle two vertices in every longest path. Similarly, every n -vertex tree has a centroid consisting of one vertex or two adjacent vertices. In the first case removal of the vertex splits the tree into subtrees of fewer than n /2 vertices.
The vertex-connectivity statement of Menger's theorem is as follows: . Let G be a finite undirected graph and x and y two nonadjacent vertices. Then the size of the minimum vertex cut for x and y (the minimum number of vertices, distinct from x and y, whose removal disconnects x and y) is equal to the maximum number of pairwise internally disjoint paths from x to y.
The distance between any two vertices in a graph is the length of the shortest path having the two vertices as its endpoints. domatic A domatic partition of a graph is a partition of the vertices into dominating sets. The domatic number of the graph is the maximum number of dominating sets in such a partition. dominating