VERTEX DECOMPOSABILITY PATH COMPLEXES OF DYNKIN GRAPHS
Abstract
Let G be a simple graph and ∆t(G) be a simplicial complex whose
facets correspond to the paths of length t(t ≥ 2) in G. Let Ln be a line graph
on vertices {x1, . . . , xn} and {xj , yj} be whisker of Ln at xj with 3 ≤ j ≤
n − 1. We give a necessary and sufficient condition that ∆t(Ln ∪ W(xj )) is
vertex decomposable, where Ln ∪ W(xj ) is called the graph obtained from
Ln by adding a whisker at vertex xj . As a consequence of our results, vertex
decomposability path complexes of Dynkin graphs are shown.
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