Remarks on the Complexity of Non-negative Signed Domination

This paper is motivated by the concept of nonnegative signed domination that was introduced by Huang, Li, and Feng in 2013 [15]. We study the non-negative signed domination problem from the theoretical point of view. For networks modeled by strongly chordal graphs and distance-hereditary graphs, we show that the non-negative signed domination problem can be solved in polynomial time. For networks modeled by bipartite planar graphs and doubly chordal graphs, however, we show that the decision problem corresponding to the non-negative signed domination problem is NP-complete. Furthermore, we show that even when restricted to bipartite planar graphs or doubly chordal graphs, the non-negative signed domination problem is not fixed parameter tractable .

[1]  Peter J. Slater,et al.  Generalized domination and efficient domination in graphs , 1996, Discret. Math..

[2]  Liming Cai,et al.  Advice Classes of Parameterized Tractability , 1997, Ann. Pure Appl. Log..

[3]  D. Rose.  Triangulated graphs and the elimination process , 1970 .

[4]  Gerard J. Chang,et al.  Algorithmic Aspects of Domination in Graphs , 1998 .

[5]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[6]  Jeremy P. Spinrad,et al.  Doubly Lexical Ordering of Dense 0 - 1 Matrices , 1993, Inf. Process. Lett..

[7]  Gen-Huey Chen,et al.  Dynamic Programming on Distance-Hereditary Graphs , 1997, ISAAC.

[8]  Lutz Volkmann,et al.  A disproof of Henning's conjecture on irredundance perfect graphs , 2002, Discret. Math..

[9]  Chuan-Min Lee,et al.  Variations of Y-dominating functions on graphs , 2008, Discret. Math..

[10]  Robert E. Tarjan,et al.  Three Partition Refinement Algorithms , 1987, SIAM J. Comput..

[11]  Ji,et al.  On the Signed Domination in Graphs , 2022 .

[12]  Wensheng Li,et al.  On Nonnegative Signed Domination in Graphs and its Algorithmic Complexity , 2013, J. Networks.

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

[14]  Martin Farber,et al.  Characterizations of strongly chordal graphs , 1983, Discret. Math..

[15]  Feodor F. Dragan,et al.  Dually Chordal Graphs , 1993, SIAM J. Discret. Math..

[16]  S. Hedetniemi,et al.  Domination in graphs : advanced topics , 1998 .

[17]  Robert E. Tarjan,et al.  Simple Linear-Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs , 1984, SIAM J. Comput..

[18]  Johannes H. Hattingh,et al.  The algorithmic complexity of signed domination in graphs , 1995, Australas. J Comb..

[19]  Alan A. Bertossi,et al.  Dominating Sets for Split and Bipartite Graphs , 1984, Inf. Process. Lett..

[20]  M. Moscarini.  Doubly chordal graphs, steiner trees, and connected domination , 1993, Networks.

[21]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[22]  Michael R. Fellows,et al.  Parameterized complexity: A framework for systematically confronting computational intractability , 1997, Contemporary Trends in Discrete Mathematics.