A graph $G=(V,E)$ is a star-$k$-PCG if there exists a weight function $w: V \rightarrow R^+$ and $k$ mutually exclusive intervals $I_1, I_2, \ldots I_k$, such that there is an edge $uv \in E$ if and only if $w(u)+w(v) \in \bigcup_i I_i$. These graphs are related to two important classes of graphs: PCGs and multithreshold graphs. It is known that for any graph $G$ there exists a $k$ such that $G$ is a star-$k$-PCG. Thus, for a given graph $G$ it is interesting to know which is the minimum $k$ such that $G$ is a star-$k$-PCG. We define this minimum $k$ as the star number of the graph, denoted by $\gamma(G)$. Here we investigate the star number of simple graph classes, such as graphs of small size, caterpillars, cycles and grids. Specifically, we determine the exact value of $\gamma(G)$ for all the graphs with at most 7 vertices. By doing so we show that the smallest graphs with star number 2 are only 4 and have exactly 5 vertices; the smallest graphs with star number 3 are only 3 and have exactly 7 vertices. Next, we provide a construction showing that the star number of caterpillars is one. Moreover, we show that the star number of cycles and two dimensional grid graphs is 2 and that the star number of $4$-dimensional grids is at least 3. Finally, we conclude with numerous open problems.

翻译：图 $G=(V,E)$ 是star-$k$-PCG，如果存在一个权函数 $w: V \rightarrow R^+$ 和 $k$ 个互不相交的区间 $I_1, I_2, \ldots, I_k$，使得边 $uv \in E$ 当且仅当 $w(u)+w(v) \in \bigcup_i I_i$。这类图与两个重要的图类相关：PCG 和多阈值图。已知对于任意图 $G$，存在一个 $k$，使得 $G$ 是star-$k$-PCG。因此，对于给定图 $G$，找出使 $G$ 为star-$k$-PCG 的最小 $k$ 值具有重要意义。我们将这个最小 $k$ 值定义为图的星号数，记为 $\gamma(G)$。本文研究了简单图类的星号数，例如小规模图、毛毛虫图、圈图和网格图。具体地，我们确定了所有顶点数不超过 7 的图的 $\gamma(G)$ 精确值。由此发现，星号数为 2 的最小图仅有 4 个，且恰好有 5 个顶点；星号数为 3 的最小图仅有 3 个，且恰好有 7 个顶点。接下来，我们给出一个构造，证明毛毛虫图的星号数为 1。此外，我们证明圈图和二维网格图的星号数为 2，而四维网格图的星号数至少为 3。最后，我们提出若干未解决问题。