15 problems

Let $(X,c_2,c_1)$ be a digital picture with no hole, and let $H_{rc}(X)$ denote its rc-convex hull. Rc-convex hull conjecture. The digital picture $(H_{rc}(X),c_2,c_1)$ is a set of…

Applicability conjecture. The approach developed in the paper is applicable to Jordan curves satisfying this condition.

A finitely presented path triangulation free group is a free group associated with a finitely presented path triangulation. A free group presentation is a presentation of a group b…

A path triangulation is a triangulation obtained from paths in a planar cell complex, and a triangulated digital image is a digital image equipped with such a triangulation. Geomet…

Let $X$ be a $c_1$-digital image, and let $dH_q(X)$ and $H_q^{c_1}(X)$ denote, respectively, its digital cubical singular homology and its $c_1$-cubical homology in degree $q$. Iso…

Let $C_m$ denote the digital cycle with $m$ vertices, and let $H_q(C_m)$ and $\bar H_q(C_m)$ be the homology theories used in the paper. Digital-cycle homology conjecture. For…

Let $X\subset \mathbb{Z}^n$ be a digital image with $c_1$-adjacency. The groups $\bar H_q(X)$ and $\bar H_q^{c_1}(X)$ are, respectively, the cubical and elementary cubical homology…

Let $X\subset \mathbb{Z}^n$ and $Y\subset \mathbb{Z}^m$ be digital images with $c_1$-adjacency, and let $f:X\to Y$ be continuous. Chain-map conjecture. The induced homomorphism … i…

Coincidence spectrum conjecture. The coincidence point spectra satisfy

Let $X$ be a contractible digital image and let $f:X\longrightarrow X$ be a continuous mapping. Here, $HFS(f,f)$ denotes the homotopic fixed point set for the pair $(f,f)$, $S(f)$…

Converse extremality conjecture. The converses of these implications are true: every maximal element of $\mathcal{J}$ contains no open points of $\mathcal{D}$, and every minimal el…

Regularity conjecture. The adjacency relation $c_k$ on ${\mathbb Z}^n$ with $k\le n$ is regular.

4-adjacency cycle conjecture. The image $X$ is non-rigid and irreducible if and only if $X$ is isomorphic to $C_n$ for some $n>4$.

Planarity conjecture. If $X$ is a non-rigid irreducible digital image that is not isomorphic to $C_n$, then the adjacency graph of $X$ is not planar.

Let $X$ be a finite digital image, viewed as a graph, and let a simple $m$-loop mean a simple loop in $X$ with $m$ vertices. Homotopy triviality conjecture. If $X$ has no simple…