Contractible transformations do not change the homology groups of graphs

Periodicity of Generalized Fibonacci-like Sequences

“…Topological properties of K are similar to topological properties of its continuous counterpart. For example, the Euler characteristic and the homology groups of a continuous and a digital Klein bottle are the same ( [7] and [8]). Consider the solution of equation (15) …”

“…A digital space G is a simple undirected graph G=(V,W) where V=(v 1 ,v 2 ,...v n ,…) is a finite or countable set of points, and W = ((v р v q ),....) is a set of edges. The induced subgraph ( )containing point v and all points adjacent to v is called the ball of point v in G. Graphs that are digital counterparts of continuous n-dimensional manifolds were studied in [5,6,7,8]. Consider the numerical solutions of the initial value problem for the wave equation on a graph G(v 1 , v 2 , … v s ).…”

Graph Theoretical Models of Discrete Spaces with Locally Non-spherical Topology

“…A digital space G is a simple undirected graph G=(V, W) where V=(v 1 , v 2 , ... v n , …) is a finite or countable set of points, and W = ((v р v q ),....) is a set of edges. Topological properties of G as a digital space in terms of adjacency, connectedness and dimensionality are completely defined by set W. Let G and v be a graph and a point of G. In ( [13], [14] For two graphs G=(X, U) and H=(Y, W) with disjoint point sets X and Y, their join G⊕H is the graph that contains G, H and edges joining every point in G with every point in H.…”

“…Contractible graphs and contractible transformations are basic elements in this approach (see [13,14]). …”

“…Contractible transformations of graphs seem to play the same role in this approach as a homotopy in algebraic topology. In papers [13,14], it was shown that contractible transformations retain the Euler characteristic and homology groups of a graph. All graphs depicted in Figure 1 are homotopy equivalent, i. e., any graph can be converted to a one-point graph G 1 by contractible transformations.…”