Planar ternary graphs, flag spheres, and Delannoy polynomials

In 2022 Kim showed when a graph $G$ is ternary (without induced cycles of length divisible by three), its independence complex $\text{Ind}(G)$ is either contractible or homotopy equivalent to a sphere. In this paper, we show that when $\text{Ind}(G)$ is homotopy equivalent to a sphere of dimension $\dim \text{Ind}(G)$, the complex is Gorenstein. Equivalently, $G$ is a $1$-well-covered graph. This answers a question by Faridi and Holleben. We then focus on the independence complexes of Gorenstein planar ternary graphs. We prove that they are boundaries of vertex decomposable simplicial polytopes. We show that the transformations among these flag spheres using edge subdivisions and contractions can be modeled by the Hasse diagram of the partition refinement poset. In addition, their $h$-polynomials are products of Delannoy polynomials and thus real-rooted. Finally, we demonstrate a way to construct nonplanar Gorenstein ($1$-well-covered) ternary graphs from planar ones.