Representing graphs as the intersection of cographs and threshold graphs

A graph $G$ is said to be the intersection of graphs $G_1,G_2,\dots ,G_k$ if $V(G)=V(G_1)=V(G_2)=\cdots =V(G_k)$ and $E(G)=E(G_1)\cap E(G_2)\cap\cdots\cap E(G_k)$\mathrm{.}Foragraph$G$,$$\mathrm{d}\mathrm{i}\mathrm{m}${COG}(G)$(resp\mathrm{.}$$\mathrm{d}\mathrm{i}\mathrm{m}${TH}(G)$) 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) ${\mathrm{d}\mathrm{i}\mathrm{m}}_{COG}(G)\le \mathrm{t}\mathrm{w}(G)+2$, (b) ${\mathrm{d}\mathrm{i}\mathrm{m}}_{TH}(G)\le \mathrm{p}\mathrm{w}(G)+1$, and © ${\mathrm{d}\mathrm{i}\mathrm{m}}_{TH}(G)\le \chi (G)\cdot \mathrm{b}\mathrm{o}\mathrm{x}(G)$, where $\mathrm{t}\mathrm{w}(G)$, $\mathrm{p}\mathrm{w}(G)$, $\chi (G)$ and $\mathrm{b}\mathrm{o}\mathrm{x}(G)$ 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 ${\mathrm{d}\mathrm{i}\mathrm{m}}_{COG}(G)$ and ${\mathrm{d}\mathrm{i}\mathrm{m}}_{TH}(G)$ when $G$ belongs to some special graph classes.