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Differential equations or difference equations on such graphs can be employed to leverage the graph's structure for tasks such as image segmentation (where the vertices represent pixels and the weighted edges encode pixel similarity based on comparisons of Moore neighborhoods or larger windows), data clustering, data classification, or ...
A graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex. In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph ...
Several algorithms based on depth-first search compute strongly connected components in linear time.. Kosaraju's algorithm uses two passes of depth-first search. The first, in the original graph, is used to choose the order in which the outer loop of the second depth-first search tests vertices for having been visited already and recursively explores them if not.
A directed graph is weakly connected (or just connected [9]) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph. A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices (x, y).
An independent set of ⌊ ⌋ vertices (where ⌊ ⌋ is the floor function) in an n-vertex triangle-free graph is easy to find: either there is a vertex with at least ⌊ ⌋ neighbors (in which case those neighbors are an independent set) or all vertices have strictly less than ⌊ ⌋ neighbors (in which case any maximal independent set must have at least ⌊ ⌋ vertices). [4]
The first few graphs in this sequence are the graph M 2 = K 2 with two vertices connected by an edge, the cycle graph M 3 = C 5, and the Grötzsch graph M 4 with 11 vertices and 20 edges. In general, the graph M i is triangle-free, (i−1)-vertex-connected, and i-chromatic. The number of vertices in M i for i ≥ 2 is 3 × 2 i−2 − 1 ...
The two edges along the cycle adjacent to any of the vertices are not written down. Let v be the vertices of the graph and describe the Hamiltonian circle along the p vertices by the edge sequence v 0 v 1, v 1 v 2, ...,v p−2 v p−1, v p−1 v 0. Halting at a vertex v i, there is one unique vertex v j at a distance d i joined by a chord with v i,
A d-claw in a graph is a set of d+1 vertices, one of which (the "center") is connected to the other d vertices, but the other d vertices are not connected to each other. A d-claw-free graph is a graph that does not have a d-claw subgraph. Consider the algorithm that starts with an empty set, and incrementally adds an arbitrary vertex to it as ...