Degree-constrained spanning tree: Difference between revisions

Revision as of 04:21, 22 February 2019

Latest revision as of 19:27, 6 February 2025

(6 intermediate revisions by 5 users not shown)

Line 1:

[[File:Degree-constrained spanning tree.png|thumb|350px|On the left, a spanning tree can be constructed where the vertex with the highest [[Degree (graph theory)|degree]] is 2 (thus, a max degree 2 tree).<br /> On the right, the central vertex must have degree at least 5 in any tree spanning this graph, so a 2 degree constrained tree cannot be constructed here.]]

In [[graph theory]], a '''degree-constrained spanning tree''' is a [[spanning tree (mathematics)|spanning tree]] where the maximum [[Degree (graph theory)|vertex degree]] is limited to a certain [[Constant (mathematics)|constant]] ''k''. The '''degree-constrained spanning tree problem''' is to determine whether a particular [[Graph (discrete mathematics)|graph]] has such a spanning tree for a particular ''k''.

==Formal definition==

Input: ''n''-node undirected graph G(V,E); positive [[integer]] ''k'' ≤< ''n''.

Question: Does G have a spanning tree in which no [[Node (computer science)|node]] has degree greater than ''k''?

==NP-completeness==

This problem is [[NP-complete]] {{harv|Garey|Johnson|1979}}. This can be shown by a [[Reduction (complexity)|reduction]] from the [[Hamiltonian path problem]]. It remains NP-complete even if ''k'' is fixed to a value ≥&nbsp;2. If the problem is defined as the degree must be ≤&nbsp;''k'', the ''k'' = 2 case of degree-confined spanning tree is the Hamiltonian path problem.

This problem is [[NP-complete]] {{harv|Garey|Johnson|1979}}. This can be shown by a reduction from the [[Hamiltonian path problem]]. It remains NP-complete even if ''k'' is fixed to a value ≥&nbsp;2. If the problem is defined as the degree must be ≤&nbsp;''k'', the ''k'' = 2 case of degree-confined spanning tree is the Hamiltonian path problem.

==Degree-constrained minimum spanning tree==

Line 19 ⟶ 20:

==Approximation Algorithm==

{{harvtxt|Fürer|Raghavachari|1994}} gavegive an approximationiterative algorithmpolynomial fortime the problemalgorithm which, ongiven anya givengraph instance<math>G</math>, eitherreturns showsa thatspanning thetree instancewith hasmaximum degree no treelarger ofthan <math>\Delta^* + 1</math>, where <math>\Delta^*</math> is the minimum possible maximum degree over all spanning trees. Thus, if <math>k or= it\Delta^*</math>, findssuch andan returnsalgorithm will either return a spanning tree of maximum degree <math>k</math> or <math>k+1</math>.

==References==