The general position problem in graphs seeks the largest set of vertices such that no three vertices lie on a common geodesic. Its counting refinement, the general position polynomial $ψ(G)$, asks for all such possible sets. In this paper, We describe general position sets for several classes of graphs and provide explicit formulas for the general position polynomials of complete multipartite graphs. We specialize to balanced complete multipartite graphs and show that for part size $r\le 4$, the polynomial $ψ(K_{r,\dots,r})$ is log-concave and unimodal for all numbers of parts, while for larger $r$, counterexamples show that these properties fail. Finally, we analyze the corona $G\circ K_1$ and prove that unimodality of $ψ(G)$ is retained for numerous natural classes (paths, edgeless graphs, combs). This contributes to an open problem, but the general case remains unknown. Our findings support the parallel between general position polynomials and classical position-type parameters, and identify balanced multipartite graphs and coronas as promising testbeds for additional research.

翻译：图论中的一般位置问题旨在寻找最大的顶点集，使得其中任意三个顶点不共处于同一条测地线上。其计数精化形式——一般位置多项式ψ(G)则枚举所有此类可能的集合。本文描述了若干图类的一般位置集，并给出了完全多部图的一般位置多项式显式公式。我们聚焦于平衡完全多部图，证明当部规模r≤4时，多项式ψ(K_{r,…,r})对所有部数均满足对数凹性与单峰性；而对于更大的r，反例表明这些性质不成立。最后，我们分析冠图G∘K₁，证明对于许多自然图类（道路图、无边图、梳齿图），ψ(G)的单峰性得以保持。这为开放问题提供了新线索，但一般情形仍悬而未决。我们的发现支持了一般位置多项式与经典位置型参数之间的平行关系，并指出平衡完全多部图和冠图可作为进一步研究的重要测试平台。