Degree-preserving trees

Abstract

We consider the degree‐preserving spanning tree (DPST) problem: Given a connected graph G, find a spanning tree T of G such that as many vertices of T as possible have the same degree in T as in G. This problem is a graph‐theoretical translation of a problem arising in the system‐theoretical context of identifiability in networks, a concept which has applications in, for example, water distribution networks and electrical networks. We show that the DPST problem is NP‐complete, even when restricted to split graphs or bipartite planar graphs, but that it can be solved in polynomial time for graphs with a bounded asteroidal number and for graphs with a bounded treewidth. For the class of interval graphs, we give a linear time algorithm. For the class of cocomparability graphs, we give an O(n4) algorithm. Furthermore, we present linear time approximation algorithms for planar graphs of a worst‐case performance ratio of 1 − ϵ for every ϵ > 0. © 2000 John Wiley & Sons, Inc.

Fixed-Parameter Tractability Results for Full-Degree Spanning Tree and Its Dual

“…FDST is motivated by applications in water distribution and electrical networks [4,8,25]. However, also biological applications concerning the interference of reaction rates in metabolic networks are conceivable [30,Chapter 8].…”

A Moderately Exponential Time Algorithm for Full Degree Spanning Tree

“…It turned out that to measure flows in all pipes, it is sufficient to find a full degree spanning tree T of the network and install flow meters (or pressure gauges) at each vertex of T that does not have full degree. We refer to [1,4,19] for a more detailed description of various applications of FDST.…”

“…Pothof and Schut [28] studied this problem first and gave a simple heuristic algorithm. The decision version of the problem, asking whether a graph has a spanning tree with at least k full degree vertices, was shown to be NP-complete by Bhatia et al [1] and by Broersma et al [4]. Bhatia et al also gave an approximation algorithm of factor O( √ n).…”

“…Lokshtanov et al [24] showed that the analog to FDST on directed graphs is W[1]-hard and that the dual can be solved in time 5.942 k n O(1) on directed graphs. Further, Broersma et al [4] have shown that…”

“…Polynomial time algorithms were also designed for strongly chordal graphs and directed path graphs [23]. However, it has been shown that the problem remains NP-hard for split graphs [4], bipartite planar graphs of maximum degree 5, and planar graphs of maximum degree 3 [6]. The goal of this paper is to study Full Degree Spanning Tree in the context of moderately exponential time algorithms, another coping strategy to deal with NP-hardness.…”