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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 ...
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 ...
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]
In the study of graph coloring problems in mathematics and computer science, a greedy coloring or sequential coloring [1] is a coloring of the vertices of a graph formed by a greedy algorithm that considers the vertices of the graph in sequence and assigns each vertex its first available color. Greedy colorings can be found in linear time, but ...
In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph.A path is called simple if it does not have any repeated vertices; the length of a path may either be measured by its number of edges, or (in weighted graphs) by the sum of the weights of its edges.
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]
Hosted by comedian Jeff Foxworthy, the original show asked adult contestants to answer questions typically found in elementary school quizzes with the help of actual fifth-graders as teammates ...
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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