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 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.