A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v, the neighbors of v that occur later than v in the ordering form a clique. Akcoglu et al  define an extension of chordal graphs whereby the neighbors of v that occur later than v in the elimination order have at most k independent vertices. We refer to such graphs as sequentially k-independent graphs. We study properties of such families of graphs, and we show that several natural classes of graphs are sequentially k-independent for small k. In particular, any intersection graph of translates of a convex object in a two dimensional plane is a sequentially 3-independent graph; furthermore, any planar graph is a sequentially 3-independent graph. For any fixed constant k, we develop simple, polynomial time approximation algorithms for sequentially k-independent graphs with respect to several well-studied NP-complete problems.
KeywordsPlanar Graph Vertex Cover Intersection Graph Chordal Graph Color Class
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