3 - Extremal Graph Properties

Summary

An interesting question in graph theory is to find whether a given graph contains a vertex induced subgraph satisfying a certain property. We shall consider the special case where the property depends only on the value of a certain parameter that can take (positive) integer values. Such properties are described as weighted properties. For a given graph G = (V,E), G′ ⊆ G means that G′ is a vertex induced subgraph of G. The generic Induced Subgraph of High Weight problem (ISHW) consists in, given a graph G = (V, E), a weighted property W on the set of graphs, and an integer κ, to decide if G contains a vertex induced subgraph H such that W(H) ≥ κ. A first concrete example of the Induced Subgraph of High Weight problem is the case when W is the minimum degree of a graph. This instance of the problem is known as the High Degree Subgraph problem (HDS). Historically, this is the first problem shown to be P-complete and approximated in parallel ([AM86] and [HS87]). Another instance of the Induced Subgraph of High Weight is the case when W is the vertex (or edge) connectivity of G. These problems are known as the High Vertex Connected Subgraph problem (HVCS) (respectively the High Edge Connected Subgraph problem (HECS)). As we shall show in Chapter 8, these problems are also P-complete for any κ ≥ 3.

Anderson and Mayr studied the High Degree Subgraph problem and found that the approximability of the problem exhibits a threshold type behavior. This behavior implies that below a certain value of the absolute performance ratio, it remains P-complete, even for fixed value κ.