How to construct polygonal region corresponding to graph? I am reading Peter Savaliev's Topology illustrated, I'm having a bit of trouble understanding what's going on in page-33.
Context:

A realization $|G|$ of graph $G$ is a subset of the Euclidean space that is the union of the following two subsets of the space:

*

*A collection of points $|N|$, one for each node $N$ in $G$, and


*A collection of paths $|E|$, one for each edge $E$ in $G$, with no intersections other than the points in $|N|$
In other words $a,b \in |N|$ are connected by a path $p \in |E|$ iiff there is an edge in $AB \in E$ where $A,B$ are the nodes corresponding to points $a,b$ as follows:
$a = |A|, b = |B|$ and $p=|AB|$
Picture: 

Later when defining a path connected graph, it seems that the geometric realization is taken as to be contained in a polygonal region:



How was the above (left picture) constructed from the definition of geometric realization?
Intuitively speaking, I sort of get it. I think that the region corresponding would be convex iff the graph is path connected at each points. The concavity comes from it not being possible to connect any two arbitrary points... but, I don't get the exact construction of how the polygon encompassing the graph was made. Could someone explain that?
 A: Regarding the picture in question, namely the drawing of the grey polygon labelled $G$, it was not "constructed from the definition of geometric realization". It is instead an artistic product of the author's imagination, a kind of cartoon even, that is intended to convey an intuitive impression to the reader.
It's actually very common for mathematicians to use pictures as an attempt to enhance intuition. But we all learn to be wary: a picture is only a cartoon of mathematical reality.
For instance, when I teach trigonometry to precalculus students I will often draw a picture: an $x$, $y$ axis; and a circle of radius $1$. But this picture is really just a cartoon. My ability to draw axes and circles is limited: my axes are never perfectly straight and thin; my circles are never perfectly round. Sometimes students get confused by this picture, but I think that most often they reach a satisfactory internal resolution of their confusions: Oh, I understand: chalk is not perfect, hand and arm muscles cannot draw perfectly, but I can get the correct idea from the drawing.
Now let me go one level deeper. When I teach topology to upper level undergraduates and graduate students, I will often draw a cartoon of a topological space $X$: using the chalk I draw an irregular roundish shape (intentionally not perfectly round), I place the label "$X$" beside that shape, and then I use the chalk to draw four or five dots inside that shape to represent points of the topological space. This picture is by no means intended to give a complete impression of $X$: it is simply a cartoon, drawn from my imagination. What I do next with that picture depends on further context. For instance, perhaps I want to depict a subspace $Y \subset X$ (I don't know if you have read yet about subspaces of topological spaces, so instead just think about a subset, if that helps). To do that, I draw a smaller irregular blob inside the $X$ blob encircling two or three of the previously drawn dots, and I then label that smaller blob with a $Y$. And, frankly, these cartoons do run a serious risk of confusing students. In a class, part of my job as a teacher of mathematics is to try to detect those confusions, and add further details or give further explanations in an attempt to dispel the confusion, until finally I can see that the students correctly understand the idea that I have tried to convey, in a preliminary fashion, with the cartoon that I drew.
Back to the picture in your post: that grey polygon is a cartoon of a graph, it is not intended as a construction of the graph itself. Here is my artistic interpretation of several different parts of that picture:

*

*It is drawn as a polygon is to invite the viewer to imagine some of the nodes and edges, namely the corners and the sides of the polygon.

*The red dots are intended to invite the viewer to imagine two more nodes.

*The inside of the polygon is colored grey to invite the reader to imagine maybe billions and trillions and quadrillions and gajillions more vertices and edges of the graph, so many that your vision could not possibly distinguish them all, they would appear as almost a homogeneous grey blur to you.

*The green curve connecting $A$ and $B$ is an invitation to imagine a long path of edges from the node $A$ to the node $B$, maybe a billion or a trillion or a quadrillion or a gajillion edges long, so long that the eye cannot possibly distinguish the individual corners along that path of edges, the whole thing appears as almost a smooth curve to you.

Now, there is one more very big hint to let you know that my artistic interpretation is correct: to the right of the grey polygon in question, the author has shown you kind of "image under the microscope" of the part of the graph inside the small rectangle. And in that microscopic image you can see: several individual nodes, including $A$; several individual edges; and in green you can see the first eleven of the bajillion edges along the path of edges from $A$ to $B$; the point $B$ itself is far, far out beyond the bounds of this microscopic image. But I'll warn you: Even that microscopic image is a cartoon.
Having said all of that, whenever you feel that a picture has misled you, here's what I recommend. If it's in a classroom setting, ask questions for clarification. If it's in a book or on a web site, and if you cannot work out what the picture is intended to mean, then you are free to just ignore that picture entirely. Hopefully you will be able to build your understanding without it.
