Line graph: Difference between revisions

Revision as of 04:14, 21 September 2020

Revision as of 02:21, 30 September 2020

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{{distinguish|path graph}}

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

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== Example ==

The following figures show a graph (left, with blue vertices) and its line graph (right, with green vertices). Each vertex of the line graph is shown labeled with the pair of endpoints of the corresponding edge in the original graph. For instance, the green vertex on the right labeled 1,3 corresponds to the edge on the left between the blue vertices 1 and 3. Green vertex 1,3 is adjacent to three other green vertices: 1,4 and 1,2 (corresponding to edges sharing the endpoint 1 in the blue graph) and 4,3 (corresponding to an edge -sharing the endpoint 3 in the blue graph).

<gallery widths="135px" heights="135px">