Linear Residuals and Gallai-Simplicial Complexes

In this paper, we give a new algebraic criterion for the {\em shellability} of (non-pure) simplicial complex $\mathrm{\Delta }$ over $[n]$, shellable in the sense of Bj"orner and Wachs \cite{BW}. We show that the spanning simplicial complex of doubly uni-cyclic graph is non-pure shellable. Moreover, we introduce the concept of Gallai-simplicial complex ${\mathrm{\Delta }}_{\mathrm{\Gamma }}(G)$ of a finite simple graph $G$. We applied the obtained criterion to discuss the shellability of Gallai simplicial complexes associated to various classes of graphs…