Partitioning perfect graphs into comparability graphs

We study how many comparability subgraphs are needed to partition the edge set of a perfect graph. We show that many classes of perfect graphs can be partitioned into (at most) two comparability subgraphs and this holds for almost all perfect graphs. On the other hand, we prove that for interval graphs an arbitrarily large number of comparability subgraphs might be necessary.

💡 Research Summary

The paper investigates the problem of partitioning the edge set of a perfect graph into comparability subgraphs. For a graph G the authors introduce two parameters: p(G), the minimum number of edge‑disjoint comparability subgraphs whose union equals E(G), and c(G), the minimum number of (not necessarily disjoint) comparability subgraphs that together cover E(G). Clearly p(G) ≥ c(G), and both equal 1 exactly when G itself is a comparability graph.

The first major result shows that for “almost all” perfect graphs both parameters are bounded by 2. “Almost all” means that as the number of vertices n tends to infinity, the proportion of labelled perfect graphs that can be partitioned into at most two comparability graphs tends to 1. The proof combines the Strong Perfect Graph Theorem with a structural theorem of Prömel and Steger on generalized split (GSP) graphs. A GSP graph is either a split graph or the complement of a split graph, i.e., its vertex set can be partitioned into a clique V₁ and a collection of vertex‑disjoint cliques V₂, or the complement has this form. In either case the edges can be split into a comparability graph that is a union of cliques and a bipartite graph between V₁ and V₂, giving a partition into two comparability graphs. Since almost all Berge graphs are GSP, the same holds for almost all perfect graphs.

The second set of results establishes the same bound for several well‑studied perfect‑graph families. For line graphs of bipartite graphs (the class LBIP) the edges can be labelled according to which side of the original bipartite graph they touch; each label class forms a union of disjoint cliques, yielding two comparability subgraphs. For unit‑interval graphs the authors present a greedy algorithm that groups intervals by the rightmost endpoint of the leftmost interval; intervals in the same group form a disjoint union of cliques, while edges between different groups form a bipartite graph, again giving a two‑graph partition. For co‑triangulated graphs (disjointness graphs of subtrees of a tree) they construct two partial orders on the set of subtrees: one based on the distance of the subtree’s root to a fixed tree root, and a second lexicographic order on a coding of the root. The comparability graphs of these two orders partition the disjointness graph, proving p(G) ≤ 2 for this class as well.

The third and most striking contribution concerns interval graphs, a subclass of perfect graphs where the bound 2 does not hold in general. Let Gₙ be the interval graph formed by all closed intervals with integer endpoints among