Let G be a connected graph on n vertices. A spanning tree T of G is called an independence tree, if the set of end vertices of T (vertices with degree one in T ) is an independent set in G. If G has an independence tree, then α t (G) denotes the maximum number of end vertices of an independence tree of G. We show that determining α t of a graph is an NP-hard problem. We give the following analogue of a well-known result due to Chvátal and Erdös. If α t (G) ≤ κ(G) − 1, then G is hamiltonian. We show that the condition is sharp. An I ≤k -tree of G is an independence tree of G with at most k end vertices or a Hamilton cycle of G. We prove the following two generalizations of a theorem of Ore. If G has an independence tree T with k end vertices such that two end vertices of T have degree sum at least n − k + 2 in G, then G has an I ≤k−1 -tree. If the degree sum of each pair of nonadjacent vertices of G is at least n − k + 1, then G has an I ≤k -tree. Finally, we prove the following analogue of a closure theorem due to Bondy and Chvátal. If the degree sum of two nonadjacent vertices u and v of G is at least n − 1, then G has an I ≤k -tree if and only if G + uv has an I ≤k -tree (k ≥ 2). The last three results are all best possible with respect to the degree sum conditions.

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.

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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(na)), and graphs of bounded asteroidal number. Description of the Problem and Its Practical NicheAnalysis of communication or distribution networks is often concerned with finding spanning trees (or forests) of those networks fulfilling certain criteria. Also in other contexts spanning trees show up as important tools in modeling and analyzing problems. Therefore, a myriad of problems on spanning trees have been