  We recall first Gallai-simplicial complex $\Delta_{\Gamma}(G)$ associated to
Gallai graph $\Gamma(G)$ of a planar graph $G$. The Euler characteristic is a
very useful topological and homotopic invariant to classify surfaces. In
Theorems 3.2 and 3.4, we compute Euler characteristics of Gallai-simplicial
complexes associated to triangular ladder and prism graphs, respectively.
  Let $G$ be a finite simple graph on $n$ vertices of the form $n=3l+2$ or
$3l+3$. In Theorem 4.4, we prove that $G$ will be $f$-Gallai graph for the
following types of constructions of $G$.
  Type 1. When $n=3l+2$. $G=\mathbb{S}_{4l}$ is a graph consisting of two
copies of star graphs $S_{2l}$ and $S'_{2l}$ with $l\geq 2$ having $l$ common
vertices.
  Type 2. When $n=3l+3$. $G=\mathbb{S}_{4l+1}$ is a graph consisting of two
star graphs $S_{2l}$ and $S_{2l+1}$ with $l\geq 2$ having $l$ common vertices.
