Degree-Preserving Forests

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 e.g., 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. We present linear time approximation algorithms for planar graphs of worst case performance ratio 1−e for every constant e > 0. Furthermore we give exact algorithms for interval graphs (linear time), graphs of bounded treewidth (linear time), cocomparability graphs (O(n 4)), and graphs of bounded asteroidal number.

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