I am studying graphs on surfaces. I found the following definition.
A graph $\Gamma$ embedded in a surface $S$ is filling if each connected component of $|\Gamma|$ is diffeomorphic to a disc. Here, $|\Gamma|$ denotes the geometric realization of $\Gamma$.
The definition is given in Page15.
Actually, I am not getting the filling terminology. If you please help me to understand this, then it will be very helpful for me. I want to visualize the filling word. Thanks
A "filling graph" is non-standard terminology for a cellularly embedded graph, which is also called a 2-cell embedding or a map (in the sense of a geographic map, not in the sense of a function). Diffeomorphism is also a stronger condition than what is usually considered (a homeomorphism).
The definition requires three kinds of objects: a connected graph $\Gamma$, a surface $S$ and a realization (embedding) $|\Gamma|$ of $\Gamma$ on $S$. For $|\Gamma|$ to be filling, whatever you obtain by removing $|\Gamma|$ from $S$ must be a disjoint union of (spaces diffeomorphic/homeomorphic to) open discs.
Let's look at a few examples. Let $\Gamma$ be the graph given by one vertex $v$ and one loop edge $e$ going from $v$ to $v$. If $|\Gamma|$ is the realization of $\Gamma$ on $S$ such that the edge doesn't cross itself (so it's just a circle with a point on it that represents $v$), then $|\Gamma|$ is filling because $S\setminus |\Gamma|$ is a union of two open discs, the "internal" disc and the "big" outer disc.
However, if we take $S$ to be a genus 1 torus and $|\Gamma|$ to be the same realization on $S$ as above, then $|\Gamma|$ is not filling because now the "big" outer component of $S\setminus |\Gamma|$ is no longer homeomorphic to a disk. In fact, there is no way to realize $\Gamma$ on a torus such that $|\Gamma|$ is filling (this follows from the Euler-Poincare formula).
Take now $\Gamma$ to be the graph with one vertex $v$ and two loop edges based at $v$. Let now $S$ be the sphere. Then $\Gamma$ has the realization $|\Gamma|$ which looks like $\infty$. This realization is filling because if we remove $\infty$ from the sphere we get 3 connected components, each homeomorphic to an open disc.
Let now $S$ be a genus 1 torus. Then the realization of $\Gamma$ as $\infty$ is not filling because there will be one connected component that is not homeomorphic to an open disc. The shaded part in the image below is not homeomorphic to an open disk, while the two unshaded parts are homeomorphic to an open disk.
However, $\Gamma$ has a different realization, call it $|\Gamma|_2$ on a genus 1 torus which is filling. This is the following realization:
To see why $S\setminus |\Gamma|_2$ is homeomorphic to an open disk, imagine cutting $S$ first along the vertical edge. You get a tube. Now cut the tube along the horizontal edge. It unravels into an open rectangle. And an open rectangle is clearly homeomorphic to an open disk.