Elimination graphs

TLDR: It turns out that if the authors take <i>P</i> as a graph with maximum independent set size no greater than <i*k</i>, then this definition gives a natural generalization of both chordal graphs and (<i-k + 1)-claw-free graphs.

Abstract: In this article we study graphs with inductive neighborhood properties. Let P be a graph property, a graph G = (V, E) with n vertices is said to have an inductive neighborhood property with respect to P if there is an ordering of vertices v1, …, vn such that the property P holds on the induced subgraph G[N(vi)∩ Vi], where N(vi) is the neighborhood of vi and Vi = {vi, …, vn}. It turns out that if we take P as a graph with maximum independent set size no greater than k, then this definition gives a natural generalization of both chordal graphs and (k + 1)-claw-free graphs. We refer to such graphs as inductive k-independent graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are inductive k-independent for small k. In particular, any intersection graph of translates of a convex object in a two dimensional plane is an inductive 3-independent graph; furthermore, any planar graph is an inductive 3-independent graph. For any fixed constant k, we develop simple, polynomial time approximation algorithms for inductive k-independent graphs with respect to several well-studied NP-complete problems. Our generalized formulation unifies and extends several previously known results.