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Solving the Maximum Independent Set Problem for -free planar graphs

Solving the Maximum Independent Set Problem for -free planar graphs. Sarah Bleiler DIMACS REU 2005 Advisor: Dr. Vadim Lozin, RUTCOR. k. j. S i,j,k. Definitions. Planar: a graph which can be drawn in a plane with no edges crossing.

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Solving the Maximum Independent Set Problem for -free planar graphs

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

  2. k j Si,j,k 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 . 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 • H is connected • 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. Y v1 Z … Let Y denote the set of white neighbors of v1 and let Z denote the set of black vertices matched with the vertices of Y.

  10. Suppose there exist pairs of vertices of Z which share a neighborhood in Y of cardinality >2… v1 Y … Z

  11. Suppose there exist pairs of vertices of Z which share a neighborhood in Y of cardinality =2… v1 Y … Z

  12. Suppose there exist pairs of vertices of Z which share a neighborhood in Y of cardinality =1… v1 Y … Z

  13. Suppose there do not exist pairs of vertices of Z which share neighborhoods in Y… v1 Y … Z

  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? 1 1 2 1 1 1 1 1

  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. <http://mathworld.wolfram.com/MaximumIndepende ntSet.html>

  16. bleilesa@shu.edu

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