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Graph coloring [2] [3]: GT4 Graph homomorphism problem [3]: GT52 Graph partition into subgraphs of specific types (triangles, isomorphic subgraphs, Hamiltonian subgraphs, forests, perfect matchings) are known NP-complete. Partition into cliques is the same problem as coloring the complement of the given graph.
defines only one solution (), the so-called singular solution, whose graph is the envelope of the graphs of the general solutions. The singular solution is usually represented using parametric notation, as ( x ( p ) , y ( p ) ) {\displaystyle (x(p),y(p))} , where p = d y / d x {\displaystyle p=dy/dx} .
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
Solution of a travelling salesman problem: the black line shows the shortest possible loop that connects every red dot. In the theory of computational complexity, the travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the ...
On finite graphs with non-negative edge weights A* is guaranteed to terminate and is complete, i.e. it will always find a solution (a path from start to goal) if one exists. On infinite graphs with a finite branching factor and edge costs that are bounded away from zero ((,) > > for some fixed ), A* is guaranteed to terminate only if there ...
For example, sometimes it is desirable to present solutions which are less than mathematically optimal. To obtain a ranked list of less-than-optimal solutions, the optimal solution is first calculated. A single edge appearing in the optimal solution is removed from the graph, and the optimum solution to this new graph is calculated.
For example, the solution set for the above equation is a line, since a point in the solution set can be chosen by specifying the value of the parameter z. An infinite solution of higher order may describe a plane, or higher-dimensional set. Different choices for the free variables may lead to different descriptions of the same solution set ...
First, it can be false in practice. A theoretical polynomial algorithm may have extremely large constant factors or exponents, rendering it impractical. For example, the problem of deciding whether a graph G contains H as a minor, where H is fixed, can be solved in a running time of O(n 2), [25] where n is the number of vertices in G.