Domination numbers and homotopy in certain ternary graphs

TL;DR

This paper explores the topological properties of ternary graphs, showing that their independence complexes are homotopy equivalent to spheres under certain cycle restrictions, linking these to domination numbers.

Contribution

It establishes a new connection between the homotopy type of independence complexes and domination parameters in ternary graphs with specific cycle constraints.

Findings

Independence complex is homotopy equivalent to a sphere when not contractible.

Sphere dimension equals the independent domination number minus one.

Results extend to a hypergraph analogue.

Abstract

A ternary graph is a graph with no induced cycles of length $0$ modulo $3$. It was recently shown that, if the independence complex of a ternary graph is not contractible, then it is homotopy equivalent to a sphere. When a ternary graph also does not contain induced cycles of length $1$ modulo $3$, we prove that the dimension of the sphere is equal to the dimension of a minimum maximal simplex of the independence complex, or equivalently, to the value obtained by subtracting $1$ from the independent domination number of the graph. The same statement holds if we replace the independent domination number with the domination number. We also give a hypergraph analogue of the statement above.