Talk:Line graph

message: Please convert the AASCI Art graphs to images.  Thank you. —Preceding unsigned comment added by 213.211.224.201 (talk • contribs) 11:24, 15 July 2005 (UTC)
replies: text: If you could give some examples of the kind of line graphs you were looking for, we might be able to provide some links to information about them.  —Ilmari Karonen (talk) 20:22, 26 September 2007 (UTC)
thread_title: Untitled

message: Since this is a math article, I suggest putting the references in the standard mathematical format, e.g., names in first-last order.  Since I can't do that with harvtxt, I ask someone who can to fix them.  Thank you. Zaslav (talk) 03:59, 16 April 2008 (UTC)
replies: text: Re: "the standard mathematical format" for citations, please supply a reliable source for your assertion that such a thing exists. I don't believe that it does. And in any case, I think we should be aiming for a style that's more consistent across Wikipedia as a whole rather than splitting it up into incompatible fiefdoms. —David Eppstein (talk) 04:43, 16 April 2008 (UTC)
replies: text: The source is reading 95+% of published math, and guidelines like (but not limited to) the AMS book on "Math Into Type" and the format guidelines provided by journals.  This is hardly a secret.  Maybe "standard" is ambiguous; change it to "usual".  The rest of what you say is a matter of preference.  I understood the Wikipedia guidelines to suggest following the format usual in each field as a perfectly viable alternative to following the "standard" Wikipedia format.  If you disagree, please provide a source. Zaslav (talk) 18:41, 21 April 2008 (UTC)

text: Not that I think it's worth a lot of work, either way.  I simply dislike the "last-first" name format and I claim it's legitimate not to use it in Wikipedia math articles.  If you must change things, it's your lookout. Zaslav (talk) 18:46, 21 April 2008 (UTC)
thread_title:  Reference style 

message: It says the cuboctahedron is the line graph of both the cube and the octahedron but this seems to contradict the Whitney isomorphism theorem. From some quick doodling I seem to find that the edge graph of the octahedron is a cuboctahedron plus extra edges connecting opposite vertices of the square faces.
Gedkins (talk) 20:26, 5 August 2010 (UTC)
replies: text: Yes, I think you're correct. Will fix. —David Eppstein (talk) 21:09, 5 August 2010 (UTC)
thread_title:  Line graphs of convex polyhedra - wrong? 

message: "A graph G is the line graph of some other graph, if and only if it is possible to find a collection of cliques in G, partitioning the edges of G, such that each vertex of G belongs to at most two of the cliques. In order to do this, it may be necessary for some of the cliques to be single vertices." — This does not make any sense to me. I guess "at most" should read "exactly". This seems to be consistent with the rest of the article and would ensure that the reconstruction procedure (based on such a partition) results in an edge for each vertex of the line graph. The paper by Roussopoulos also uses the term "at most" to characterize line graphs, which is sufficient for characterization, but not the basis for reconstruction, I guess. --Nlskrg (talk) 00:16, 12 August 2010 (UTC)
thread_title:  Reconstruction 

message: I think the line about the linear time algorithm could be expanded into a subsection describing the algorithm, and am working on such an expansion, but am temporarily stuck at an unclear point of the paper. Here is what I have so far:
: showed how to test whether a given graph is a line graph, and if so construct a graph G such that the input is isomorphic to L(G), in linear time. Roussopoulos's algorithm depends on two equivalent characterizations of line graphs described by : their edges can be covered by cliques such that each vertex belongs to exactly two cliques, and they are the claw-free graphs in which every two odd triangles with a shared edge belong to a common clique. Here, a triangle in a graph is said to be odd if there exists a vertex outside the triangle that is adjacent to an odd number of vertices inside the triangle, and even otherwise. An odd triangle in a line graph necessarily represents three edges that share a common endpoint, because the other way of forming a triangle (a triangle in G that translates into a triangle in L(G)) cannot be odd.
:The first step of Roussopoulos' algorithm is to find a single clique in the clique cover of the supposed line graph input. To do so, the algorithm chooses an arbitrary edge e, and (by testing adjacency of its endpoints to the other vertices) finds all triangles involving e. If there are none, e is the desired clique. If there is only one triangle involving e, and the same property is true of the other two edges of the triangle, then this triangle is the desired clique. In the remaining case, the algorithm has found an edge (either e or one of the other edges of an initial triangle, which subsequently replaces e) that belongs to multiple triangles. [AT THIS POINT THE ALGORITHM COUNTS HOW MANY OF THE TRIANGLES AT e ARE ODD. HOW DOES IT DO IT?]
Will try to come back to this later. —David Eppstein (talk) 19:34, 8 November 2013 (UTC)
replies: text: Resolution of the problem: it doesn't matter. The algorithm of Degiorgi & Simon (1995) (used for the static recognition problem despite being a dynamic algorithm) is much simpler, so that's the one to describe. —David Eppstein (talk) 22:14, 8 November 2013 (UTC)
thread_title:  The algorithm