Weighted Connected Domination and Steiner Trees in Distance-hereditary Graphs

Abstract Distance-hereditary graphs are graphs in which every two vertices have the same distance in every connected induced subgraph containing them. This paper studies distance-hereditary graphs from an algorithmic viewpoint. In particular, we present linear-time algorithms for finding a minimum weighted connected dominating set and a minimum vertex-weighted Steiner tree in a distance-hereditary graph. Both problems are N P -complete in general graphs.

[1]  Lorna Stewart,et al.  A Linear Recognition Algorithm for Cographs , 1985, SIAM J. Comput..

[2]  Martin Farber,et al.  Domination, independent domination, and duality in strongly chordal graphs , 1984, Discret. Appl. Math..

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

[4]  Marina Moscarini,et al.  Distance-Hereditary Graphs, Steiner Trees, and Connected Domination , 1988, SIAM J. Comput..

[5]  Lorna Stewart,et al.  Dominating sets in perfect graphs , 1991, Discret. Math..

[6]  Peter J. Slater,et al.  R-Domination in Graphs , 1976, J. ACM.

[7]  David S. Johnson.  The NP-Completeness Column: An Ongoing Guide , 1986, J. Algorithms.

[8]  Derek G. Corneil,et al.  Complement reducible graphs , 1981, Discret. Appl. Math..

[9]  Ortrud R. Oellermann,et al.  Steiner Distance-Hereditary Graphs , 1994, SIAM J. Discret. Math..

[10]  Peter L. Hammer,et al.  Completely separable graphs , 1990, Discret. Appl. Math..

[11]  A. Brandstädt,et al.  A linear-time algorithm for connected r-domination and Steiner tree on distance-hereditary graphs , 1998 .

[12]  Hans-Jürgen Bandelt,et al.  Distance-hereditary graphs , 1986, J. Comb. Theory, Ser. B.

[13]  Gerard J. Chang.  Labeling algorithms for domination problems in sun-free chordal graphs , 1988, Discret. Appl. Math..

[14]  C. Pandu Rangan,et al.  Weighted Independent Perfect Domination on Cocomparability Graphs , 1993, Discret. Appl. Math..

[15]  Martin Farber,et al.  Steiner trees, connected domination and strongly chordal graphs , 1985, Networks.

[16]  Andreas Brandstädt,et al.  The NP-Completeness of Steiner Tree and Dominating Set for Chordal Bipartite Graphs , 1987, Theor. Comput. Sci..

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

[18]  Antonio Sassano,et al.  The Steiner Tree Problem and Homogeneous Sets , 1988, MFCS.

[19]  Gérard Cornuéjols,et al.  The traveling salesman problem on a graph and some related integer polyhedra , 1985, Math. Program..

[20]  Charles J. Colbourn,et al.  Permutation graphs: Connected domination and Steiner trees , 1991, Discret. Math..

[21]  E. Howorka.  A CHARACTERIZATION OF DISTANCE-HEREDITARY GRAPHS , 1977 .

[22]  Yehoshua Perl,et al.  Clustering and domination in perfect graphs , 1984, Discret. Appl. Math..

[23]  Arie Segev,et al.  The node-weighted steiner tree problem , 1987, Networks.

[24]  Stefan Arnborg,et al.  Linear time algorithms for NP-hard problems restricted to partial k-trees , 1989, Discret. Appl. Math..