Contractible independence complexes of trees

TL;DR

This paper characterizes when the independence complex of a tree is contractible, linking it to specific reduction sequences and polynomial evaluations, thus advancing understanding of tree structures in combinatorial topology.

Contribution

It provides a novel characterization of contractible independence complexes of trees through truncation moves and polynomial evaluations.

Findings

Independence complex of a tree is contractible iff it reduces to a path with length congruent to 1 mod 3.

Characterizes trees where the independence polynomial at -1 equals 1 or -1.

Establishes a connection between topological properties and polynomial evaluations 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 \( -1 \).