VERTEX DECOMPOSABILITY PATH COMPLEXES OF DYNKIN GRAPHS - AJDANI, SEYED MOHAMMAD (IJRDO -JOURNAL OF MATHEMATICS, 2017)
Let G be a simple graph and ∆t(G) be a simplicial complex whosefacets correspond to the paths of length t(t ≥ 2) in G. Let Ln be a line graphon 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 )) isvertex decomposable, where Ln ∪ W(xj ) is called the graph obtained fromLn by adding a whisker at vertex xj . As a consequence of our results, vertexdecomposability path complexes of Dynkin graphs are shown.