Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces

For $X insert ignore into journalissuearticles values(\subset R^n);$, assume the subspace $insert ignore into journalissuearticles values(X, E_X^n);$ induced by the $n$-dimensional Euclidean topological space $insert ignore into journalissuearticles values(R^n, E^n);$. Let $Z$ be the set of integers. Khalimsky topology on $Z$, denoted by  $insert ignore into journalissuearticles values(Z, \kappa);$, is generated by the set $\{\{2m-1, 2m, 2m+1\}\,\vert\, m \in {Z}\}$ as a subbase. Besides, Khalimsky topology on  $Z^n, n \in N$, denoted by $insert ignore into journalissuearticles values(Z^n, \kappa^n);$, is a product topology induced by $insert ignore into journalissuearticles values({Z}, \kappa);$. Proceeding with a digitization of $insert ignore into journalissuearticles values(X, E_X^n);$ in terms of the Khalimsky insert ignore into journalissuearticles values($K$-, for short); topology, we obtain a $K$-digitized space in ${Z}^n$, denoted by $D_Kinsert ignore into journalissuearticles values(X); insert ignore into journalissuearticles values(\subset {Z}^n$);, which is a $K$-topological space. Considering further $D_Kinsert ignore into journalissuearticles values(X);$ with $K$-adjacency, we obtain a topological graph related to the $K$-topology insert ignore into journalissuearticles values(a $KA$-space for short); denoted by $D_{KA}insert ignore into journalissuearticles values(X);$ insert ignore into journalissuearticles values(see an algorithm in Section 3);. Motivated by an $A$-homotopy between $A$-maps for $KA$-spaces,  the present paper establishes a new homotopy, called an $LA$-homotopy, which is suitable for studying homotopic properties of both $insert ignore into journalissuearticles values(X, E_X^n);$ and $D_{KA}insert ignore into journalissuearticles values(X);$ because a homotopy for Euclidean topological spaces has some limitations of digitizing $insert ignore into journalissuearticles values(X, E_X^n);$. The goal of the paper is to study some relationships among an ordinary homotopy equivalence for spaces $insert ignore into journalissuearticles values(X, E_X^n);$, an $LA$-homotopy equivalence for spaces $insert ignore into journalissuearticles values(X, E_X^n);$, and an $A$-homotopy equivalence for $KA$-spaces $D_{KA}insert ignore into journalissuearticles values(X);$. Finally, we classify  $KA$-spaces insert ignore into journalissuearticles values(resp. $insert ignore into journalissuearticles values(X, E_X^n););$ via an $A$-homotopy equivalence insert ignore into journalissuearticles values(resp. an $LA$-homotopy equivalence);. This approach can facilitate studies of applied topology, approximation theory and digital geometry.