A graph $G$ is said to be the intersection of graphs $G_1,G_2,\ldots,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$. For a graph $G$, $\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$) denotes the minimum number of cographs (resp. threshold graphs) whose intersection gives $G$. We present several new bounds on these parameters for general graphs as well as some special classes of graphs. It is shown that for any graph $G$: (a) $\mathrm{dim}_{COG}(G)\leq\mathrm{tw}(G)+2$, (b) $\mathrm{dim}_{TH}(G)\leq\mathrm{pw}(G)+1$, and (c) $\mathrm{dim}_{TH}(G)\leq\chi(G)\cdot\mathrm{box}(G)$, where $\mathrm{tw}(G)$, $\mathrm{pw}(G)$, $\chi(G)$ and $\mathrm{box}(G)$ denote respectively the treewidth, pathwidth, chromatic number and boxicity of the graph $G$. We also derive the exact values for these parameters for cycles and show that every forest is the intersection of two cographs. These results allow us to derive improved bounds on $\mathrm{dim}_{COG}(G)$ and $\mathrm{dim}_{TH}(G)$ when $G$ belongs to some special graph classes.

中文翻译：

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将图表示为 cograph 和阈值图的交集

如果 $V(G)=V(G_1)=V(G_2)=\cdots=V(G_k)$ 和 $，则称图 $G$ 是图 $G_1,G_2,\ldots,G_k$ 的交集E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$。对于图 $G$，$\mathrm{dim}_{COG}(G)$ (resp. $\mathrm{dim}_{TH}(G)$) 表示 cographs 的最小数量（resp. threshold graphs ) 其交点给出 $G$。我们为一般图以及一些特殊类别的图提供了这些参数的几个新界限。结果表明，对于任何图 $G$： (a) $\mathrm{dim}_{COG}(G)\leq\mathrm{tw}(G)+2$, (b) $\mathrm{dim} _{TH}(G)\leq\mathrm{pw}(G)+1$，和 (c) $\mathrm{dim}_{TH}(G)\leq\chi(G)\cdot\mathrm{ box}(G)$，其中 $\mathrm{tw}(G)$、$\mathrm{pw}(G)$、$\chi(G)$ 和 $\mathrm{box}(G)$ 分别表示图 $G$ 的树宽、路径宽度、色数和方正度。我们还推导出这些循环参数的确切值，并表明每个森林都是两个 cograph 的交集。当 $G$ 属于某些特殊图类时，这些结果使我们能够推导出 $\mathrm{dim}_{COG}(G)$ 和 $\mathrm{dim}_{TH}(G)$ 的改进边界。