1. Solving the Maximum Independent Set Problem for -free planar graphs Sarah Bleiler DIMACS REU 2005 Advisor: Dr. Vadim Lozin, RUTCOR

2. Definitions Planar: a graph which can be drawn in a plane with no edges crossing. -free: A graph which contains no connected components of the form. j k S i,j,k i

3. Theorem 1 [Lozin, Mosca] Let X be a subclass of planar graphs defined by a finite set F of forbidden induced planar graphs. If, then the maximum independent set problem is NP-hard in the class X.

4. Augmenting Graphs Let S be any independent set in G. Label V(S) as white and V(G-S) as black. A bipartite graph H=(W,B,E) of G is said to be augmenting for S if:

5. Minimal Augmenting Graphs If H=(W,B,E) is a minimal augmenting graph for an independent set S, then i.H is connected ii. iii.For every subset

6. Our Problem Find a solution to the MIS problem in -free planar graphs. Procedure: (a) find a complete list of augmenting graphs in the class under consideration (b) develop polynomial-time algorithms for detecting all augmenting graphs in the class

7. Find augmenting graphs of bounded degree Theorem 2 [Lozin, Milanič]: For any positive integers n and d, there are only finitely many -free minimal augmenting graphs of maximal degree at most d, different from strips and bracelets.

8. Find augmenting graphs of unbounded degree Conjecture: In the class of -free planar graphs, there are no augmenting graphs with a vertex of unbounded degree. Hall’s Theorem: A bipartite graph H with bipartitions B and W has a perfect matching iff for all subsets A of W.

9. Let Y denote the set of white neighbors of v 1 and let Z denote the set of black vertices matched with the vertices of Y. … Y Z v1v1

10. Suppose there exist pairs of vertices of Z which share a neighborhood in Y of cardinality >2… … Y Z v1v1

11. Suppose there exist pairs of vertices of Z which share a neighborhood in Y of cardinality =2… … Y Z v1v1

12. Suppose there exist pairs of vertices of Z which share a neighborhood in Y of cardinality =1… … Y Z v1v1

13. Suppose there do not exist pairs of vertices of Z which share neighborhoods in Y… … Y Z v1v1

14. An algorithm to find augmenting strips? We know how to find augmenting chains Characterize the “twins” of the augmenting strips. Reduction to claw-free weighted graphs? 2111 1 111

15. References [1] A. Hertz, V.V. Lozin, The Maximum Independent Set Problem and Augmenting Graphs. Graph Theory and Combinatorial Optimization, 1:1-32, 2005. [2] V.V.Lozin, M. Milanič, On Finding Augmenting Graphs. [3] V.V. Lozin, R. Mosca, Maximum Independent Sets in Planar Graphs. RUTCOR Research Report 40, 2004. [4] Eric W. Weisstein. "Maximum Independent Set." From Mathworld--A Wolfram Web Resource.

16. bleilesa@shu.edu