Dominating 2-broadcast in graphs: complexity, bounds and extremal graphs

Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating $2$-broadcasts along with the associated parameter, the dominating $2$-broadcast number. We prove that computing the dominating $2$-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desired.

[1]  Haitze J. Broersma.  Existence of Δλ-cycles and Δλ-paths , 1988, J. Graph Theory.

[2]  Michael S. Jacobson,et al.  Minimum Degree and Dominating Paths , 2017, J. Graph Theory.

[3]  Fred R. McMorris,et al.  Graphs with only caterpillars as spanning trees , 2003, Discret. Math..

[4]  Pinar Heggernes,et al.  Optimal broadcast domination in polynomial time , 2006, Discret. Math..

[5]  Nader Jafari Rad,et al.  Limited Dominating Broadcast in graphs , 2013, Discret. Math. Algorithms Appl..

[6]  Claude Berge,et al.  The theory of graphs and its applications , 1962 .

[7]  G. Dirac.  Some Theorems on Abstract Graphs , 1952 .

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

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

[10]  F. Harary,et al.  On Eulerian and Hamiltonian Graphs and Line Graphs , 1965, Canadian Mathematical Bulletin.

[11]  F. Harary,et al.  The theory of graphs and its applications , 1963 .

[12]  Henk Jan Veldman,et al.  Existence of dominating cycles and paths , 1983, Discret. Math..

[13]  Christina M. Mynhardt,et al.  A class of trees with equal broadcast and domination numbers , 2013, Australas. J Comb..

[14]  Eugene L. Lawler,et al.  Linear-Time Computation of Optimal Subgraphs of Decomposable Graphs , 1987, J. Algorithms.

[15]  Sarada Herke,et al.  Broadcasts and domination in trees , 2011, Discret. Math..

[16]  C. L. Liu,et al.  Introduction to Combinatorial Mathematics. , 1971 .