Definition talk:Walk (Graph Theory)

0-length walk
May a walk have length 0? I.e., may a walk be formed by only one vertex and zero edges? I don't think so, but it is not clear in the definition.

But if it were allowed, the definition of "connected vertices" could be simplified.


 * No, nothing to stop it. I see what you about simplifying that definition, but clarity could suffer if we're not careful. --prime mover 01:15, 17 May 2012 (EDT)


 * Awodey requires these to be paths as well (they play the rôle of identity in the free category on a digraph). Would it be necessary to separate the defs for digraphs and undigraphs? --Lord_Farin 08:32, 17 August 2012 (UTC)


 * It appears that the whole graph theory section is way overdue for a refactorisation process. As apparently there is currently only one source (which, as I discovered on an internet journey, was deemed by p.m to be mediocre at best in the rigour compartment) it will be a challenge to regulate everything without introducing idiosyncrasies. Unfortunately, I have no sources treating the subject in any depth (and my ebook collection does not show any books with interest in digraphs) so that it is likely that we have to rely on ourselves here. --Lord_Farin 09:50, 17 August 2012 (UTC)


 * Some web search yielded the freely accessible book on digraphs by Bang-Jensen and Gutin. It may help us. --Lord_Farin 09:53, 17 August 2012 (UTC)


 * Until we get our hands on a properly rigorous and scholarly work on graph theory, we're sort of in danger of guessing our way round the subject, arranging things the way we think they ought to be rather than as the discipline has itself grown up into. Unfortunately the only graph theory materials I have access to are the aforementioned Chartrand (it's not truly bad, it's just a little bit thin on content) and my original graph theory notes, the latter of which I have not formally documented as sources because they're buried at the bottom of somewhere inaccessible at the moment.


 * If you have the urge to rewrite the G.T. stuff then feel free, but the usual caveats apply. --prime mover 12:17, 17 August 2012 (UTC)


 * A zero length walk is just a single vertex. Such walk is well-defined and I also think the definition should be updated, in Length section. --Mloskot (talk) 13:02, 8 January 2013 (UTC)


 * We know. If you read the above discussion, we've determined that most of the graph theory section is up for review. We haven't located a person with the prerequisites time, skill and sources to do the job for us, though. If we start patching up stuff as we encounter trouble, the result may well be worse than what it is now. Therefore, we've chosen to, well, practically leave this part of the site alone until a new or regular contributor is located that has or has attained said three elements for a sound rewriting job. Thanks for noticing, though. --Lord_Farin (talk) 13:13, 8 January 2013 (UTC)

Alternative definitions
This definition of walk is very heavy. It appears that in the case of simple graphs and digraphs, it is more common to just use sequences of vertices. The power of this definition only seems to come into play when dealing with multigraphs. How should we refactor this? --Dfeuer (talk) 05:33, 29 May 2013 (UTC)


 * By separating out the subsections into their separate pages.


 * If you can find a rigorous definition of walk using simpler language (not MathWorld, it's carelessly inaccurate in many places) then feel free to copy it here, but I recommend (a) express carefully the limitations of such a cut-down definition, and (b) write a page demonstrating (1) that the full definition degenerates to the cut-down definition for the simple case and (2) why the cut-down version does not apply to multigraphs and loop graphs. --prime mover (talk) 06:10, 29 May 2013 (UTC)


 * It's not even an issue for loop graphs, where you'd just have $(x)$, $(x,x)$, $(x,x,x)$, etc. as the walks from $x$ to $x$. It's only an issue for multigraphs, multidigraphs, and loop versions of those. I found one so-so non-Mathworld source. I'll try for something better. --Dfeuer (talk) 06:58, 29 May 2013 (UTC)


 * On the other hand if, as you say, you don't know much about graph theory and don't have any sources for it, WTF are you doing working on it? --prime mover (talk) 18:16, 29 May 2013 (UTC)