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Variations of the maximum leaf spanning tree problem for bipartite graphs

Variations of the maximum leaf spanning tree problem for bipartite graphs. P.C. Li and M. Toulouse Information Processing Letters 97 (2006) 129-132 2006/03/14. Outline. The maximum leaf spanning tree for bipartite graph is NP-complete

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Variations of the maximum leaf spanning tree problem for bipartite graphs

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  1. Variations of the maximum leaf spanning tree problem for bipartite graphs P.C. Li and M. Toulouse Information Processing Letters 97 (2006) 129-132 2006/03/14

  2. Outline • The maximum leaf spanning tree for bipartite graph is NP-complete • The maximum leaf spanning tree for bipartite graph of maximum degree 4 is NP-complete • The maximum leaf spanning tree for planar bipartite graph of maximum degree 4 is NP-complete

  3. Maximum Leaf Spanning Tree(MLST) • Problem 1.1 Let G=(V,E) be a connected graph and let K < |V| be a positive integer. We are asked whether G contains a spanning tree T consisting of least K vertices of degree 1. • This problem is known to be NP-complete in 1979. In fact, it remains NP-complete for regular graph of degree 3(in 1988) as well as for planar graphs of maximum degree 4(in 1979) • This problem has applications in the area of communication networks

  4. S X Y MLST for Bipartite Graph • Problem 1.2 Given a connected graph G=(V,E) with partite sets X and Y and a positive number K|X|, we are asked the question of whether there is a spanning tree TG of G such that the number of leaves in TG belonging to X is greater than or equal to K • Theorem 1.3 Let G=(V,E) be a connected bipartite graph with partite sets X and Y. Let K|X| be a positive integer. Then there is a spanning tree T of G with at least K leaves in X if and only if there is a set SX such that • |X\S|K(i.e. |S||X|-K) and • the induced subgraph SY of G is connected • We will show that Problem 1.2 is NP-complete using Theorem 1.3

  5. S X YA MLST for Bipartite Graph is NP-complete • Theorem 2,1 Problem 1.2 is NP-complete • Proof. Consider an instance of the set-covering problem given by a collection  of subsets of the finite set A=c and a positive integer c||. Let K=|X|-c. Therefore,  contains a cover for A of size c or less if and only if there exists a set SX such that |X\S|K, and the induced subgraph SY of G is connected

  6. Exact cover by 3-sets(X3C) Problem • Problem 2.2 Given a finite X with |X|=3q and a collection  of 3-element subsets of X, we are asked the question of whether there is a sub-collection of  that partitions X • The X3C problem remains NP-complete if no element of X occurs in more than three subsets of 

  7. MLST for Bipartite Graph of Maximum Degree 4 is NP-complete • Problem 2.3 Given a connected bipartite graph G=(V,E) of maximum degree 4 with partite sets X and Y and a positive number K|X|, we are asked the question of whether there is a spanning tree TG of G such that the number of leaves in TG belonging to X is greater than or equal to K • Theorem 2.4 Problem 2.3 is NP-complete • Proof. Let (X,) be an instance of the X3C problem with ||=p, |X|=3q, pq, and no element of X occurs in more than three members of . We will construct a bipartite graph G=(AB,E) with maximum degree 4, such that (X,) has an exact 3-cover if and only if G contains a spanning tree with at least p-q leaves in B

  8. A B MLST for Bipartite Graph of Maximum Degree 4 is NP-complete • Proof of Theorem 2.4 Cont. We begin by building a rooted tree T* (from the bottom up) of depth log2p. T* log2p G

  9. MLST for Bipartite Graph of Maximum Degree 4 is NP-complete • The key observation here is that none of the white nodes (those in B) can be leaf nodes of a spanning tree, unless they are members of . • We can show that (X,) has an exact 3-cover if and onlt if G contains a spanning tree with at least K=p-q leaves in B

  10. MLST for Planar Bipartite Graph of Maximum Degree 4 is NP-complete • Theorem 2.5 1-in-3 satisfiability remains NP-complete if • Every variable appears in exactly 3 clauses • Negations do not occur in any clauses, and • The bipartite graph formed by joining a variable and a clause if and only if the variable appears in the clause, is planar • It can easily be seen that an instance of the restricted 1-in-3 satisfiability problem given by the conditions in Theorem 2.5m can be reduced to an instance of the X3C problem (X,) by associating X with the clauses and  with the variables of the satisfiability problem instance • Theorem 2.6 Problem 1.2 remains NP-complete for planar bipartite graph of maximum degree 4

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