Representing Graphs as the Intersection of Cographs and Threshold Graphs

Abstract

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 $G$, $\dim_{COG}(G)$ (resp. $\dim_{TH}(G)$) 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) $\dim_{COG}(G)\leqslant\mathrm{tw}(G)+2$, (b) $\dim_{TH}(G)\leqslant\mathrm{pw}(G)+1$, and (c) $\dim_{TH}(G)\leqslant\chi(G)\cdot\mathrm{box}(G)$, 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 $\dim_{COG}(G)$ and $\dim_{TH}(G)$ when $G$ belongs to some special graph classes.