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A graph with chromatic number is -vertex-critical if each of its vertices is a critical element. Critical graphs are the minimal members in terms of chromatic number, which is a very important measure in graph theory. Some properties of a -critical graph with vertices and edges: has only one component. is finite (this is the De Bruijn–Erdős ...
A critical value is the image under f of a critical point. These concepts may be visualized through the graph of f: at a critical point, the graph has a horizontal tangent if one can be assigned at all. Notice how, for a differentiable function, critical point is the same as stationary point.
A factor-critical graph, together with perfect matchings of the subgraphs formed by removing one of its vertices. In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph [1] [2]) is a graph with n vertices in which every induced subgraph of n − 1 vertices has a perfect matching. (A perfect matching in a ...
This can only occur when the graph has an odd number of vertices, and such a matching must be maximum. In the above figure, part (c) shows a near-perfect matching. If, for every vertex in a graph, there is a near-perfect matching that omits only that vertex, the graph is also called factor-critical.
In graph theory, Brooks' theorem states a relationship between the maximum degree of a graph and its chromatic number. According to the theorem, in a connected graph in which every vertex has at most Δ neighbors, the vertices can be colored with only Δ colors, except for two cases, complete graphs and cycle graphs of odd length, which require ...
For every pair of vertices in a connected graph, there exists at least one shortest path between the vertices, that is, there exists at least one path such that either the number of edges that the path passes through (for unweighted graphs) or the sum of the weights of the edges (for weighted graphs) is minimized. The betweenness centrality for ...
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The solutions of this equation are the x-values of the critical points and are given, using the quadratic formula, by =. The sign of the expression Δ 0 = b 2 – 3ac inside the square root determines the number of critical points. If it is positive, then there are two critical points, one is a local maximum, and the other is a local minimum.