摘要
如果存在一个权重函数（w：和 k 个互斥区间（I_1, I_2, \ldots I_k\ ），这样，当且仅当（w(u)+w(v)在 \bigcup _i I_i\）时，存在一条边（uv 在 E\ 中）。这些图与两类重要的图有关：成对相容图（PCG）和多阈值图。众所周知，对于任何图 G，都存在一个 k，使得 G 是星-k-PCG。因此，对于给定的图 G，我们有兴趣知道哪一个 k 是最小的，使得 G 是星形-k-PCG。我们把这个最小 k 定义为图的星形数，用 \(\gamma (G)\) 表示。在此，我们将研究简单图类的星形数，如小尺寸图、毛毛虫图、循环图和网格图。具体来说，我们确定了所有顶点最多为 7 个的图的\(\gamma (G)\) 的精确值。通过这样做，我们证明了星号为 2 的最小图形只有 4 个，并且正好有 5 个顶点；星号为 3 的最小图形只有 3 个，并且正好有 7 个顶点。接下来，我们提供了一种构造，表明毛毛虫的星号是 1。此外，我们还证明了循环图和二维网格图的星号是 2，而四维网格图的星号至少是 3。最后，我们提出了许多有待解决的问题。A graph \(G=(V,E)\) is a star-k-pairwise compatibility graph (star-k-PCG) if there exists a weight function \(w: V \rightarrow \mathbb {R}^+\) 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: pairwise compatibility 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.
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