# The maximum independent set problem on layered graphs

The independent set on a graph $G=(V,E)$ is a subset of $V$ such that no two vertices in the subset have an edge between them. The maximum independent set problem on $G$ seeks to identify an independent set with maximum cardinality, i.e. maximum independent set or MIS. The maximum independent set problem on a general graph is known to be NP-complete. On certain classes of graphs MIS can be computed in polynomial time. Such algorithms are known for bipartite graphs, chordal graphs, cycle graphs, comparability graphs, claw-free graphs, interval graphs and circular arc graphs. On trees, the weighted version of this problem can be solved in linear time. In this article we introduce a new type of graph called a layered graph and show that if the number of vertices in a layer is $O(\log \mid V \mid)$ then the maximum independent set can be computed in polynomial time.

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