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In graph theory, a deletion-contraction formula / recursion is any formula of the following recursive form: = + (/). Here G is a graph, f is a function on graphs, e is any edge of G, G \ e denotes edge deletion, and G / e denotes contraction. Tutte refers to such a function as a W-function. [1]
If s and t are specified vertices of the graph G, then an s – t cut is a cut in which s belongs to the set S and t belongs to the set T. In an unweighted undirected graph, the size or weight of a cut is the number of edges crossing the cut. In a weighted graph, the value or weight is defined by the sum of the weights of the edges crossing the ...
The problems of finding a Hamiltonian path and a Hamiltonian cycle can be related as follows: In one direction, the Hamiltonian path problem for graph G can be related to the Hamiltonian cycle problem in a graph H obtained from G by adding a new universal vertex x, connecting x to all vertices of G. Thus, finding a Hamiltonian path cannot be ...
Matching (graph theory) MaxDDBS; Maximal independent set; Maximum agreement subtree problem; Maximum common edge subgraph; Maximum common induced subgraph; Maximum cut; Maximum flow problem; Maximum weight matching; Metric k-center; Minimum k-cut; Mixed Chinese postman problem; Multi-trials technique
The "pearls" of the title include theorems, proofs, problems, and examples in graph theory.The book has ten chapters; after an introductory chapter on basic definitions, the remaining chapters material on graph coloring; Hamiltonian cycles and Euler tours; extremal graph theory; subgraph counting problems including connections to permutations, derangements, and Cayley's formula; graph ...
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 graph theory, the metric dimension of a graph G is the minimum cardinality of a subset S of vertices such that all other vertices are uniquely determined by their distances to the vertices in S. Finding the metric dimension of a graph is an NP-hard problem; the decision version, determining whether the metric dimension is less than a given ...
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 ...