Variations of the maximum leaf spanning tree problem

Variations of the maximum leaf spanning tree problem for bipartite graphs P. C. Li and M. Toulouse Information Processing Letters 97 (2006) 129132 2006/03/14 1

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 2

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 3 communication networks

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 whethere 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 |XS| K(i. e. |S| |X|-K) and the induced subgraph S Y of G is connected S We will show that Problem 1. 2 is NP-complete using Theorem 1. 3 4 X Y

MLST for Bipartite Graph is NPcomplete 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 |XS| K, and the induced subgraph S Y of G is connected S X Y A 5

Exact cover by 3 -sets(X 3 C) Problem 2. 2 Given a finite X with |X|=3 q and a collection of 3 -element subsets of X, we are asked the question of whethere is a sub-collection of that partitions X The X 3 C problem remains NP-complete if no element of X occurs in more than three subsets of 6

MLST for Bipartite Graph of Maximum Degree 4 is NPcomplete 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 whethere 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 X 3 C problem with | |=p, |X|=3 q, 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 7

MLST for Bipartite Graph of Maximum Degree 4 is NPcomplete Proof of Theorem 2. 4 Cont. We begin by building a rooted tree T* (from the bottom up) of depth log 2 p. A B log 2 p T* G 8

MLST for Bipartite Graph of Maximum Degree 4 is NPcomplete 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 9

MLST for Planar Bipartite Graph of Maximum Degree 4 is NPcomplete 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. 5 m can be reduced to an instance of the X 3 C 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 10