On star-k-PCGs: exploring class boundaries for small k values

A graph G=(V,E) is a star-k-pairwise compatibility graph (star-k-PCG) if there exists a weight function w:V→R+ and k mutually exclusive intervals I1,I2,…Ik, such that there is an edge uv∈E if and only if w(u)+w(v)∈⋃iIi. 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 γ(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 γ(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.