This one is courtesy of frequent commenter, wst. It’s really a PoSWW, but the ideas may be sufficiently unfamiliar for it to be worth encouraging WitCHlike discussion. It comes from the Cambridge textbook *General Mathematics 1&2.*

### UPDATE (10/08/22)

By request, we’ve added an excerpt from the follow-up section of the text, on planar graphs.

### UPDATE (02/09/22)

Frequent commenter SRK has pointed out the following question from the Network module of the 2017 Further Exam 1. We *think* the question is ok enough, but there is definitely room for debate. (VCAA’s intended answer can be found here.)

Equivalent? Definitely not.

Isomorphic? The labelling of vertices is problematic. In both examples.

A lot of words that will confuse General 1&2 students? Probably.

Hmmm. The graphs in the first row are all representations of the same thing, though. Did I misunderstand your comment?

Aside from the obvious craziness of whatever bastardised version of ‘isomorphism’ they’re using…. Even if they wanted to treat it somewhat properly, you’d think explaining what ‘corresponding vertices’ is supposed to mean would make sense.

And language like “in graphical terms” just feels ripe for additional confusion. Is that Graphical in the ‘pictorial sense’? Graphical in the graph-theory sense? Graphical in the graph-means-a-plot sense? I feel like this embodies the entire mistreatment here…

Thanks, Tau. Yeah, after posting I realised it really is a WitCH. Even on its own fantasyland terms, the text is hilariously bad.

Hmmm…

The first example isn’t particularly a good example for demonstrating graph isomorphism, considering that the vertex sets are the same {A,B,C,D}. I’ll mark this by my definition of graph isomorphism which is:

There is a bijection between the vertex sets such that for , then .

In this case, it seems to be a shuffling around of the graph. Admittedly, that could still be a graph isomorphism, but if you’re abstracting away details to the actual vertex set G = (V,E), then the first example(s) are all the same graph and are isomorphic to each other, but it’s a trivial isomorphism (and an automorphism at that).

Then to the non-example, I understand they re-used the letters for the three graphs to indicate a correspondence, but… by my definition, there’d still be an isomorphism for example 1 and 3, just that D->A, C->B and so on. The way they reuse the same letters is confusing and distracts from the idea of graph isomorphism. The definition they have given is bad, because there’s a focus on the way they look, not elucidating on the information a graph contains.

Yep. Of course, there is a point to noting that graphs that look different may have the same graph properties, even without label switching. But they don’t get to steal words to make the point.

When I taught this subject, I found that many definitions in this book were inadequate as indicated by the other correspondents. Also the amount of time devoted to graph theory was quite short; one could barely get beyond the litany of definitions. I wondered why graph theory was in the the syllabus at all for General Mathematics 1-2.

So everyone could ignore the fact that the students had learned nothing in the preceding eleven years.

This is more a comment / question on the teaching of this (part of this) topic in VCE.

What is the point of including graph isomorphisms? As far as I can tell, what we really want to emphasise to students is that the way the graph is drawn on the page (ie. the relative positions of the vertices, the shape / length of the edges) is irrelevant. Surely we can do that without mentioning isomorphisms (and the inevitable confusions that result, as evidenced by the textbook’s treatment). Am I missing something?

Very good question. I checked the Cambridge Further text. The discussion there is different, with no wrong example, but with the same don’t-change-labels message, and with the incorrect definition of “isomorphism” implicit. The one exercise then screws it up.

I also checked a few Further exams, and it seems the concept of isomorphism is very seldom tested. I found one question, I think from 2019, in which the graphs included no labels and the non-isomorphicness was clear.

Thank you for posting this. It’s interesting to see what people thought. I taught the class today, and explained how you could draw the same graph different ways; and how isomorphism was a broader idea, where the names of the vertices might be different.

I was a bit surprised by the response when I said that the textbook is not quite right, so far as I would say. One student said “oh yeah, I know.” No one was surprised. The students seemed savvier about the fallibility of textbooks than I was at their age. I was kind of expecting them not to believe me, but it seemed normal to them.

ETA: since people asked, it seems the main reason they learn about it being possible to draw the same graph different ways is to learn about planar graphs (and understand they might be drawn with intersecting edges but still be planar) so they can learn Euler’s formula. I’m not sure why they extend this to isomorphic graphs, other than the allure of technical language.

Of coursethe students know the textbooks are garbage. They are only unclear on whether they are permitted to declare the texts are garbage. Which of course they are.In my short experience, most students rarely read or study the mathematics textbook. They tend to use it (the e-version) to solve the problems that the teacher has set, or complete a test or quiz.

I heard today that the ministers of education in Australia are meeting tomorrow (12 Aug) to solve the problems confronting education in our schools.

Very good news. I was getting sick of blogging.

Marty – I think minsters and mathematicians have quite different definitions of “solve”.

Pity.

Along those lines WST, the text’s presentation of planar graphs is awful, (Perhaps if Marty is willing, he could edit the initial post with a snip of p. 425 from the same book, for commenters who don’t have it handy). On p. 425, the book displays two drawings of the same (planar) graph. In diagram 1, labelled “Graph 1”, the graph is drawn with intersecting edges, and is described as “non-planar as drawn”. Whereas in diagram 2, labelled “Graph 2”, the graph is drawn without intersecting edges and is described as the “planar form of Graph 1”. To make matters worse, the text above states that Graph 1 and Graph 2 are isomorphic, which might suggest to the novice that a non-planar graph can be isomorphic to a planar graph.

Added, as requested.

Isomorphic is DIFFERENT to equivalent.

Isomorphic has a SPECIFIC MEANING and, from what I have seen (OK, I haven’t been to the source, but can infer from the excerpts) HAS NOT BEEN DEFINED and seems to be used in a very haphazard fashion.

There is a bit more WiTCH in this passage though, in my opinion, regarding the phrase “contain the same information”. More on than when I’m sober.

I notice their use of the weasel word “equivalent”, which in VCE-speak means, I believe: “sort of the same by some definition we are not going to give”. The point is that “isomorphism” in graph theory has a very precise definition, and nothing to do with labels. Either they should give that precise definition, or not talk about it at all. And what on earth does “contain the same information” mean? This is a mix-up of non-definitions that looks like it was almost designed to be confusing. How depressing.

It’s what happens when you have writers who don’t understand what they’re writing about.

So… were all the people who do know what they’re talking about too busy?

“equivalent” is a phrase Marty (and others) have taken issue with many, many times. I’m not convinced VCAA (or 95% of my colleagues, historically) actually know the difference between “equivalent” and “equal”.

It is an interesting point, in some ways.

And not in many others.

One could just use a textbook on graph theory;

Click to access wilsongraph.pdf

BTW, is now Kaliningrad and part of Russia; it has been in the news lately; during WW2 all the bridges were bombed and only 5 have been rebuilt; an opportunity to create a new problem for your students; the history of Prussia is fascinating; I recommend “Iron Kingdom: The Rise and Downfall of Prussia, 1600–1947”; well-written by Sir Christopher Clark, an Australian stuck in Cambridge.

I genuinely LOVE this book Terry. Good to see I’m not the only one.

Only problem is I feel Wilson goes way beyond where Specialist Mathematics will dare.

Guess we will wait and see.

Sure; but a teacher could at least base the definitions on Wilson

The problem is as I see it that they conflate a visual representation of a graph with the mathematical object. The definition of a graph has, strictly, nothing to do with the way it may be drawn in the plane. It is not that “graph 1”, “graph 2” and “graph 3” are all “isomorphic”. They are in fact visual representations of the same graph.

The confusion continues below: under some definition of isomorphic, “graphs” 1 and 2 (actually representations of the same graph) are actually isomorphic to “graph” 3. (“Sai” said this above.)

In the second excerpt on non-planar graphs, they again show their confusion about what a graph is. A graph is planar if it has a planar representation in the plane. Hence both diagrams represent a planar graph.

Gavin is right. A graph is either planar or it’s not. Any graph can be drawn with squiggly lines that cross each other as many times as you like. However, if the graph can be drawn with no edge crossings then it’s planar. If you want to distinguish between different sorts of drawings then you might want to talk about a planar or non-planar “embedding”, or sometimes (more confusingly) a “plane” or “non-plane” graph.

But the muckiness above from the VCE is atrocious.

To be clear, not “from the VCE” (or the VCAA), but the from the writers of the textbook.

I’ve updated the post with a related question pointed out by SRK.

The update is interesting, potentially problematic if the notion of “different way” is not clear to the student.

I think here it is pretty obvious what the question wants the student to do… so maybe it is ok?

yeah, maybe. It was kind of my feeling, too. But I can see reasonable arguments for three different answers, which ain’t great.

I didn’t mention this in my initial contact with Marty about this question, but I think this is even worse: from the examiners report, 26% chose C, 20% chose D and 29% chose E, and the examiners report says that this “suggests that simple counting errors contributed to the low success of this question”,

Well, my guess is that the examiners are correct, that the students understood the question as intended, and they counted poorly. But it’s only a guess. Of course, the alternative suggestion, that the examiners stuffed up, doesn’t rate a mention.