The complexity of the locally connected spanning tree problem

A locally connected spanning tree T of a graph G is a spanning tree of G with the following property: for every vertex, its neighbourhood in T induces a connected subgraph in G. The existence of such a spanning tree in a network ensures, in case of site and line failures, effective communication amongst operative sites as long as these failures are isolated.We prove that the problem of determining whether a graph contains a locally connected spanning tree is NP-complete, even when input graphs are restricted to planar graphs or split graphs. On the other hand, we obtain a linear-time algorithm for finding a locally connected spanning tree in a directed path graph, and a linear-time algorithm for adding fewest edges to a graph to make a given spanning tree of the graph a locally connected spanning tree of the augmented graph.

[1]  A. Brandstädt,et al.  Graph Classes: A Survey , 1987 .

[2]  Marshall Wayne Bern.  Network design problems: steiner trees and spanning k -trees , 1987 .

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  David S. Johnson,et al.  The Planar Hamiltonian Circuit Problem is NP-Complete , 1976, SIAM J. Comput..

[5]  D. West.  Introduction to Graph Theory , 1995 .

[6]  Arthur M. Farley,et al.  Networks immune to isolated failures , 1981, Networks.

[7]  M. Golumbic.  Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57) , 2004 .

[8]  Robert E. Tarjan,et al.  Fast Algorithms for Finding Nearest Common Ancestors , 1984, SIAM J. Comput..

[9]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[10]  Fanica Gavril,et al.  A recognition algorithm for the intersection graphs of directed paths in directed trees , 1975, Discret. Math..

[11]  Leizhen Cai,et al.  On Spanning 2-trees in a Graph , 1997, Discret. Appl. Math..

[12]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[13]  Paul Frederick Dietz.  Intersection graph algorithms , 1984 .

[14]  M. Golumbic.  Algorithmic graph theory and perfect graphs , 1980 .

[15]  Charles J. Colbourn,et al.  Steiner trees, partial 2-trees, and minimum IFI networks , 1983, Networks.

[16]  Leizhen Cai,et al.  On the SPANNING k-Tree Problem , 1993, Discret. Appl. Math..

[17]  F. Samarotto.  David Neumeyer and Susan Tepping. A Guide to Schenkerian Analysis. Englewood Cliffs, NJ: Prentice-Hall, 1992 , 1993 .