1 Variations of the maximum leaf spanning tree problem for bipartite graphs P.C. Li and M. Toulouse Information Processing Letters 97 (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 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 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 T G of G such that the number of leaves in T G 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 S XY

5 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 X  YAYA S

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 T G of G such that the number of leaves in T G 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 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  log 2 p .  log2p  T* G AA BB

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 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