1. 1 Complexities of some interesting problems on spanning trees M Sohel Rahman King’s College, London M Kaykobad KHU, NSU and BUET

2. 2 Abstract Complexity issues of some interesting spanning tree problems by imposing various constraints and restrictions on graph parameters. Introduce a new notion of “set version” of a problem by replacing bounds by a set of that cardinality. Maximum leaf spanning tree is one such example

3. 3 Problems under consideration Problem 1.1(Degree constrained spanning tree): Given a connected graph G=(V,E) and a positive integer K<|V|, we are asked the question whether there is a spanning tree of G such that no vertex in T has degree larger than K. Theorem 1.2: degree constrained spanning tree problem is NP-Complete.

4. 4 Problems (contd.) Problem 1.3 (maximum Leaf Spanning Tree Problem): Given a connected graph G=(V,E) and a positive integer K<|V|, we are asked the question whether there is a spanning tree of G such that K or more vertices in T have degree 1. Theorem 1.4 Maximum Leaf Spanning Tree Problem is NP-Complete.

5. 5 New Problems We denote by N G (x)- the set of vertices adjacent to vertex x, d G (x) its cardinality. Subgraph of G induced by a set S of vertices is denoted by Π G ={v|v is a leaf in G} Matching M of G from A to B ⊆ V none of A or B has degree more than 1

6. 6 New problems and results Problem 2.1 (Minimum Leaf Spanning Tree): Given a connected graph G=(V,E) and a positive integer K<|V| we are asked the question whether there is a spanning tree T of G such that K or less vertices have degree 1. Theorem 2.2 Minimum Leaf Spanning Tree Problem is NP-Complete.

7. 7 New Problems and Results(contd.) Problem 2.3 (Restricted-Leaf-in-Subgraph Spanning Tree Problem): Given G=(V,E) be a connected graph, X a vertex subset of G and a positive integer K<|X|, we are asked the question whether there is a spanning tree T G such that number of leaves in T G belonging to X is less than or equal to K.

8. 8 New problems and Results(contd.) Theorem 2.4 Restricted-Leaf-in-Subgraph Spanning Tree Problem is NP-Complete. Proof: If X=V then it is Minimum Leaf Spanning Tree Problem. Hence it is NP- Complete. Now we consider a variant of Maximum Leaf Spanning Tree for Bipartite Graphs.

9. 9 New Problems and Results(contd.) Problem 2.5(variant of Maximum Leaf Spanning Tree for Bipartite Graphs) Let G be a connected bipartite graph with partite sets X and Y. Given a positive integer K<=|X| we are asked the question whether there is a spanning tree T G in G such that the number of leaves in T G belonging to X is greater than or equal to K.

10. 10 New Problems and Results(contd.) Theorem 2.6. Let G be a connected bipartite graph with partite sets X and Y and suppose K is a positive integer such that K =K and is connected.

11. 11 New Problems and Results(contd.) Theorem 2.7 Let G be a connected bipartite graph with partite sets X and Y and suppose K is a positive integer such that K =K b) is connected c) for any subset S’ ⊆ S |N G (S’)|>=|S’|+1

12. 12 New Problems and Results(contd.) Problems of set version Problem 3.1 (Set Version of Maximum Leaf spanning Tree problem) Given a connected graph G=(V,E) and X ⊆ V, we are asked the question whether there is a spanning tree T such that X ⊆ П T, where П T ={v|v is a leaf of T}

13. 13 New Problems and Results(contd.) Theorem 3.2 Let G=(V,E) be a connected graph, X ⊆ V and Y=V-X. Then there exists a spanning tree T such that X ⊆ П T, if and only if both of the following conditions hold true: 1) is connected, 2) Every X-node has an adjacent node in Y.

14. 14 New Problems and Results(contd.) Theorem 3.3 Set version of the Maximum Leaf Spanning Tree problem is polynomially solvable. Problem 3.4 (Set version of Problem 2.5) Let G be a connected bipartite graph with partite sets X and Y and X 1 ⊆ X. we are asked the question whether there is a spanning tree T G in G such that X 1 ⊆ П T, where П T ={v|v is a leaf of T}

15. 15 New Problems and Results(contd.) Theorem 3.4 Problem 3.4 is polynomially solvable. Problem 3.6( Set version of Minimum Leaf Spanning Tree problem): Given a connected graph G=(V,E) and X ⊆ V, we are asked the question whether there is a spanning tree T such that П T ⊆ X, where П T ={v|v is a leaf of T}

16. 16 New Problems and Results(contd.) Theorem 3.7 Set version of Minimum Leaf Spanning Tree Problem is NP-Complete. References EW Dijkstra, Self-stabilizing systems in spite of distributed control, ACM 17(1974) 643-644 MR Garey, DS Johnson, Computers and Intractability, Freeman, New York, 1979 P Hall, Representation of subsets, J London Math Soc 10(1935) M Sohel Rahman, M Kaykobad, Complexities of some interesting problems on spanning trees, Information processing Letters 94(2005)93-97

17. 17 THANK YOU VERY MUCH