Hypercubes and Isometric Words Based on Swap and Mismatch Distance

The hypercube of dimension n is the graph whose vertices are $$2^n$$ binary words length n, and there an edge between two them if they have Hamming distance 1. We consider edit based on swaps mismatches, to which we refer as tilde-distance, define tilde-hypercube with edges linking at tilde-distance Then, introduce study some isometric subgraphs obtained by using special called tilde-isometric words. keep only that avoid a given word factor. An infinite family described; respect but not distance. In case 11, subgraph tilde-Fibonacci cube, generalization classical Fibonacci cube. cube can be recursively defined; same holds for number their edges. This allows asymptotic estimation in comparison total tilde-hypercube.