Line graph: Difference between revisions

Revision as of 22:52, 17 March 2022

Revision as of 03:56, 7 July 2022

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{{distinguish|path graph}}

In the [[mathematics|mathematical]] discipline of [[graph theory]], the '''line graph''' of an [[undirected graph]] ''{{mvar|G''}} is another graph {{math|L(''G'')}} that represents the adjacencies between [[edge (graph theory)|edges]] of ''{{mvar|G''}}. {{math|L(''G'')}} is constructed in the following way: for each edge in ''{{mvar|G''}}, make a vertex in {{math|L(''G'')}}; for every two edges in ''{{mvar|G''}} that have a vertex in common, make an edge between their corresponding vertices in {{math|L(''G'')}}.

The name line graph comes from a paper by {{harvtxt|Harary|Norman|1960}} although both {{harvtxt|Whitney|1932}} and {{harvtxt|Krausz|1943}} used the construction before this.{{sfnp|Hemminger|Beineke|1978|p=273}} Other terms used for the line graph include the '''covering graph''', the '''derivative''', the '''edge-to-vertex dual''', the '''conjugate''', the '''representative graph''', and the '''θ-obrazom''',{{sfnp|Hemminger|Beineke|1978|p=273}} as well as the '''edge graph''', the '''interchange graph''', the '''adjoint graph''', and the '''derived graph'''.<ref name="h72-71">{{harvtxt|Harary|1972}}, p. 71.</ref>

{{harvs|authorlink=Hassler Whitney|first=Hassler|last=Whitney|year=1932|txt}} proved that with one exceptional case the structure of a [[connected graph]] ''{{mvar|G''}} can be recovered completely from its line graph.<ref name="whitney"/> Many other properties of line graphs follow by translating the properties of the underlying graph from vertices into edges, and by Whitney's theorem the same translation can also be done in the other direction. Line graphs are [[claw-free graph|claw-free]], and the line graphs of [[bipartite graph]]s are [[perfect graph|perfect]]. Line graphs are characterized by nine [[forbidden graph characterization|forbidden subgraphs]] and can be recognized in [[linear time]].

Various extensions of the concept of a line graph have been studied, including line graphs of line graphs, line graphs of multigraphs, [[Line graph of a hypergraph|line graphs of hypergraphs]], and line graphs of weighted graphs.