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Deciding whether the metric dimension of a tree is at most a given integer can be done in linear time [10] Other linear-time algorithms exist for cographs, [5] chain graphs, [11] and cactus block graphs [12] (a class including both cactus graphs and block graphs). The problem may be solved in polynomial time on outerplanar graphs. [4]
The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 was at the origin of another branch of graph theory, extremal graph theory. The four color problem remained unsolved for more than a century. In 1969 Heinrich Heesch published a method for solving the problem using computers. [29]
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-task assignment.
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. A related problem is to find a partition that is optimal terms ...
If the graph is empty, we go to the final step 5 below. Otherwise, Wernicke's Theorem tells us that S 5 is nonempty. Pop v off S 5 , delete it from the graph, and let v 1 , v 2 , v 3 , v 4 , v 5 be the former neighbors of v in clockwise planar order, where v 1 is the neighbor of degree at most 6.
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Shortest path (A, C, E, D, F), blue, between vertices A and F in the weighted directed graph. In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized.
A subdivision of a graph is a graph formed by subdividing its edges into paths of one or more edges. Kuratowski's theorem states that a finite graph G {\displaystyle G} is planar if it is not possible to subdivide the edges of K 5 {\displaystyle K_{5}} or K 3 , 3 {\displaystyle K_{3,3}} , and then possibly add additional edges and vertices, to ...