Representing graphs as the intersection of cographs and threshold graphs

TL;DR

This paper investigates how graphs can be represented as intersections of cographs and threshold graphs, providing new bounds and exact values for these parameters across various graph classes.

Contribution

It introduces new bounds on the intersection dimensions of graphs using cographs and threshold graphs, and characterizes these parameters for specific graph classes.

Findings

- For any graph G, dim_COG(G) ≤ tw(G) + 2. - For any graph G, dim_TH(G) ≤ pw(G) + 1. - For any graph G, dim_TH(G) ≤ χ(G) · box(G). - Exact values for cycles and forests regarding these parameters. - Forests are the intersection of two cographs.

The paper establishes bounds relating intersection dimensions to treewidth, pathwidth, chromatic number, and boxicity.

It provides exact intersection dimensions for cycles and forests.

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 $G$, $\mathrm{dim}_{COG}(G)$ (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.