A polygonal complex $K$ is said to be geometrically 2-dimensional if the topological space it defines is a surface (boundaries are allowed). It is said to be $C$-quasi-1-dimensional (for some $C>0$) if there exists a graph $G$ and a simplicial mapping $f:K_m\rightarrow G$ such that for any vertex $v\in G$ the preimage $f^{-1}(v)$ includes at most $C$ vertices of $K$.
By polygonal complex I mean a generalization of simplicial 2-complex in which 2-cells are allowed to be any simple polyogon and not just a triangle
A wild conjecture would be the following:
Let $K$ be a 2-dimensional polygonal complex, and suppose that every vertex $v$ is of distance at most $R$ from the boundary of $K$ (i.e there exists a path on the 1-skeleton of $K$ of length at most $R$ which starts at the vertex $v$ and ends at a vertex which is in the boundary of $K$). Then $K$ is $C$-quasi-1-dimensional for some constant $C$.
By saying that $C$ is a constant I mean that it is not dependent in $K$ (as long as $K$ satisfies the condition of vertices being of distance at most $R$ from the boundary).
If I must guess, I would say that that this conjecture is false. However I do tend to believe that it can be proven with some other demands on $K$.
There are some conditions on $K$ which for my concern can be assumed. There are other conditions which I believe could be vital for a solution and thus are assumed as well (though if any of them can be omitted it is preferred):
- An upper bound on the degrees: the degree of each vertex of $K$ (i.e in its 1-skeleton) is either $2$ or $4$. The degree of each polygon is at most $D$.
- An upper bound on the genus: the genus of $K$ is at most some constant $\chi_{max}$
- A lower bound of "holes sizes": the boundary of $K$ consists of cycles of length above some constant $m$.
I shall emphisize that the above $C$ can indeed be dependent on $R$,$D$,$m$,$\chi_{max}$ and even exponentially.
The main strategy that I tried was to construct a graph $G'$ whose vertices are located at holes of $K$ and an edge is connected if those vertices are of distance at most $\approx 2R$ from eachother. Then I took $G$ to be the dual graph of $G'$ and defined the map $f$ which send each vertex of $G$ to the face of $G'$ which it belongs to. It is indeed true that the preimage is bounded because the faces of $G'$ have a bounded diameter, however it is not necessarily the case that $f$ is simplicial.
