What does "maximal number of closed cutting curves that do not disconnect the graph into multiple components" mean?

Question

What does "maximal number of closed cutting curves that do not disconnect the graph into multiple components" mean?

Particularly, what does closed cutting curves mean?

Answer 1

The phrase refers to the topological genus of a surface... informally, the number of "holes" in the surface.

Consider a sphere (genus $0$). Draw any closed cutting curve (a loop) on the surface of the sphere. That "cut" will separate the points on the sphere inside the loop from those on the outside of the loop. Such a curve "disconnects" the graph representing points on the sphere. This is true no matter which closed loop you draw on the sphere. Thus the "maximal" number having the property in the phrase is $0$... which is also the genus of a sphere. (The same holds for any surface of genus $0$, such as a cube, an octohedron, a complicated sculpture without holes, ...)

Now consider a donut (genus $1$). You can draw a closed loop encircling the "tube" of the donut and you will not "disconnect" the graph representing the points on the donut. (Of course, if you drew a small closed loop on the surface, you could separate points on the donut, but the statement refers to the maximal number of such closed loops.) Once you cut the donut in this way, the resulting figure has genus $0$... and you cannot make another such closed loop cut and keep the figure connected. Thus the maximum number of such loops you can create (keeping the figure connected) is $1$.

Now consider two donuts joined in the obvious way (genus $2$). Now you can draw two separate closed loops and yet keep the surface (graph) connected.

And so forth.