Some Results on Packing Graphs in their Complements

A supergraph \(H\) of a graph \(G\) is called tree-covered if \(H – E(G)\) consists of exactly \(|V(G)|\) vertex-disjoint trees, with each tree having exactly one point in common with \(G\). In this paper, we show that if a graph \(G\) can be packed in its complement and if \(H\) is a tree-covered supergraph of \(G\), then \(G\) itself is self-packing unless \(H\) happens to be a member of a specified class of graphs. This is a generalization of earlier results that almost all trees and unicyclic graphs can be packed in their complements.