Complexities of some interesting problems on spanning trees

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Complexities of some interesting problems on
spanning trees

• M Sohel Rahman
• Kings College, London
• M Kaykobad
• KHU, NSU and BUET

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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

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Problems under consideration

• Problem 1.1(Degree constrained spanning tree)
  Given a connected graph G(V,E) and a positive
  integer KltV, 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.

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Problems (contd.)

• Problem 1.3 (maximum Leaf Spanning Tree Problem)
  Given a connected graph G(V,E) and a positive
  integer KltV, 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.

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New Problems

• We denote by NG(x)- the set of vertices adjacent
  to vertex x, dG(x) its cardinality.
• Subgraph of G induced by a set S of vertices is
  denoted by ltSgt
• ?Gvv is a leaf in G
• Matching M of G from A to B?V none of A or B has
  degree more than 1

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New problems and results

• Problem 2.1 (Minimum Leaf Spanning Tree) Given a
  connected graph G(V,E) and a positive integer
  KltV 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.

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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 KltX, we are asked the question whether
  there is a spanning tree TG such that number of
  leaves in TG belonging to X is less than or equal
  to K.

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New problems and Results(contd.)

• Theorem 2.4 Restricted-Leaf-in-Subgraph Spanning
  Tree Problem is NP-Complete.
• Proof If XV 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.

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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 KltX we are asked the
  question whether there is a spanning tree TG in G
  such that the number of leaves in TG belonging to
  X is greater than or equal to K.

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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 KltX. Then there is
  a spanning tree T in G such that the number of
  leaves in T belonging to X is grater than or
  equal to K iff there is a set S?X such that
  X-SgtK and ltS?Ygt is connected.

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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 KltX. Then there is
  a spanning tree T in G such that the number of
  leaves in T belonging to X is greater than or
  equal to K iff there is a set S?X such that all
  the followings hold true a) X-SgtK b) ltS?Ygt is
  connected c) for any subset S?S NG(S)gtS1

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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??T, where ?Tvv is
  a leaf of T

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New Problems and Results(contd.)

• Theorem 3.2 Let G(V,E) be a connected graph, X?V
  and YV-X. Then there exists a spanning tree T
  such that X??T, if and only if both of the
  following conditions hold true
• 1) ltYgt is connected,
• 2) Every X-node has an adjacent node in Y.

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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 X1 ?X. we are asked the question
  whether there is a spanning tree TG in G such
  that X1 ? ?T , where ?Tvv is a leaf of T

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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 ?T ? X, where
  ?Tvv is a leaf of T

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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

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