Contractible independence complexes of trees

Abstract

We show that the independence complex of a tree is contractible if and only if it can be reduced to a path \( P_n \) with \( n \equiv 1 \pmod{3} \) by a sequence of truncation moves at branching points. As a consequence of our method, we also characterize the trees for which the independence polynomial evaluated at \( -1 \) is equal to \( 1 \) or \( -1 \).

We are grateful to Professor Alexander Engström and Thiago Holleben for pointing out relevant work on the homotopy of the independence complexes of graphs