Representing Graphs as the Intersection of Cographs and Threshold Graphs - Chacko, Daphna; Francis, Mathew C. (The Electronic Journal of Combinatorics, 2021)
A graph G is said to be the intersection of graphs G_1,G_2,…,G_k if V(G)=V(G_1)=V(G_2)=…=V(G_k) and E(G)=E(G_1)E(G_2)…E(G_k). For a graph G, _COG(G) (resp. _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) _COG(G)tw(G)+2, (b) _TH(G)pw(G)+1, and (c) _TH(G)χ(G)·box(G), where tw(G), pw(G), χ(G) and 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 _COG(G) and _TH(G) when G belongs to some special graph classes.