Sequential Elimination Graphs
نویسندگان
چکیده
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 kindependent 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.
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