@MISC{Ye_sequentialelimination, author = {Yuli Ye and Allan Borodin}, title = {Sequential Elimination Graphs }, year = {} }

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Abstract

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 [2] 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. Clearly this extension of chordal graphs also extends the class of (k+1)-claw-free 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 NPcomplete problems based on this k-sequentially independent ordering. Our generalized formulation unifies and extends several previously known results. We also consider other classes of sequential elimination graphs.