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This graph becomes disconnected when the right-most node in the gray area on the left is removed This graph becomes disconnected when the dashed edge is removed.. In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more ...
The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [18] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...
Put another way, a bipartite graph is a graph with no odd cycles; equivalently, it is a graph that may be properly colored with two colors. Bipartite graphs are often written G = (U,V,E) where U and V are the subsets of vertices of each color. However, unless the graph is connected, it may not have a unique 2-coloring. biregular
The only graphs that are connected and have minimum rank one are the complete graphs. [4] A path graph P n on n vertices has minimum rank n − 1. The only n-vertex graphs with minimum rank n − 1 are the path graphs. [5] A cycle graph C n on n vertices has minimum rank n − 2. [6] Let be a 2-connected graph.
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
A graph can only contain a perfect matching when the graph has an even number of vertices. A near-perfect matching is one in which exactly one vertex is unmatched. Clearly, a graph can only contain a near-perfect matching when the graph has an odd number of vertices, and near-perfect matchings are maximum matchings. In the above figure, part (c ...
The graph [] that remains after removing from is an induced forest (resp. an induced directed acyclic graph in the case of directed graphs). Thus, finding a minimum FVS in a graph is equivalent to finding a maximum induced forest (resp. maximum induced directed acyclic graph in the case of directed graphs).
In particular, every graph property expressible as a first-order sentence can be tested in linear time for the graphs of bounded expansion. These are the graphs in which all shallow minors are sparse graphs, with a ratio of edges to vertices bounded by a function of the depth of the minor. Even more generally, first-order model checking can be ...