M450_2 - 1 Introduction to Graph Theory MATH 450 Lecture 2...

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1Introduction to Graph TheoryMATH 450Lecture 22. Paths and Cycles2.1. ConnectivityDefinition 2.1.Given a graphG, awalkinGis a finite sequence of edges of the form01121,,,mmv vv vvvKalso denoted by0121mmvvvvvK,In which any two consecutive edges are adjacent or identical. We callv0theinitial vertexandvmthefinal vertex of the walk.The number of edges in a walk is called itslength.Fig. 2.1The walkvwxyzzywhas length 7.Definition 2.2.A walk in which all the edges are distinct is atrail. If, in addition, the vertices01,,,mvvvKare distinct (except, possibly,0mvv) then the trial is apath.Definition 2.3.A walk, trail or path isclosedif0mvv.Definition 2.4.A closed path with at least one edge is acycle.vwxyz
2On the Fig. 2vwxyzzxis a trail,vwxyzis a path,vwxyzxvis a closed trail,vwxyvis a closed cycle.Note:A loop is a cycle of length 1; a pair of multiple edges a cycle of length 2.Definition 2.5.A cycle of length 3 (such asvwxv) is called atriangle.

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Term
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Graph Theory, Planar graph, Glossary of graph theory, Graph theory objects, Eulerian path, Hamiltonian path

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