# The Connected Domination Number of Grids

11 Feb 2021-pp 247-258

Abstract: Closed form expressions for the domination number of an \(n \times m\) grid have attracted significant attention, and an exact expression has been obtained in 2011 [7]. In this paper, we present our results on obtaining new lower bounds on the connected domination number of an \(n \times m\) grid. The problem has been solved for grids with up to 4 rows and with 6 rows and the best currently known lower bound for arbitrary m, n is Open image in new window [11]. Fujie [4] came up with a general construction for a connected dominating set of an \(n \times m\) grid. In this paper, we investigate whether this construction is indeed optimum. We prove a new lower bound of Open image in new window for arbitrary \(m,n \ge 4\).

Topics: Domination analysis (58%), Image (category theory) (53%)

##### Citations

More filters

•

Abstract: Given an undirected simple graph, a subset of the vertices of the graph is a {\em dominating set} if every vertex not in the subset is adjacent to at least one vertex in the subset. A subset of the vertices of the graph is a {\em connected dominating set} if the subset is a dominating set and the subgraph induced by the subset is connected. In this paper, we determine the minimum cardinality of a connected dominating set, called the {\em connected domination number}, of an $m \times n$ grid graph for any $m$ and $n$.

##### References

More filters

•

01 Jan 1979-

Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.

39,992 citations

••

[...]

Brent N. Clark

^{1}, Brent N. Clark^{2}, Charles J. Colbourn^{1}, Charles J. Colbourn^{2}+2 more•Institutions (2)TL;DR: It is shown that many standard graph theoretic problems remain NP-complete on unit disks, including coloring, independent set, domination, independent domination, and connected domination; NP-completeness for the domination problem is shown to hold even for grid graphs, a subclass of unit disk graphs.

Abstract: Unit disk graphs are the intersection graphs of equal sized circles in the plane: they provide a graph-theoretic model for broadcast networks (cellular networks) and for some problems in computational geometry. We show that many standard graph theoretic problems remain NP-complete on unit disk graphs, including coloring, independent set, domination, independent domination, and connected domination; NP-completeness for the domination problem is shown to hold even for grid graphs, a subclass of unit disk graphs. In contrast, we give a polynomial time algorithm for finding cliques when the geometric representation (circles in the plane) is provided.

1,454 citations

••

01 Aug 1999-

Abstract: Efficient routing among a set of mobile hosts (also called nodes) is one of the most important functions in ad-hoc wireless networks. Routing based on a connected dominating set is a frequently used approach, where the searching space for a route is reduced to nodes in the set. A set is dominating if all the nodes in the system are either in the set or neighbors of nodes in the set. In this paper, we propose a simple and efficient distributed algorithm for calculating connected dominating set in ad-hoc wireless networks, where connections of nodes are determined by their geographical distances. Our simulation results show that the proposed approach outperforms a classical algorithm. Our approach can be potentially used in designing efficient routing algorithms based on a connected dominating set.

1,181 citations

••

TL;DR: Using these results, it is able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness forPlanar generalized geography.

Abstract: We define the set of planar boolean formulae, and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is NP-complete. Using these results, we are able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness for planar generalized geography.The NP-completeness of planar node cover and planar Hamiltonian circuit and line were first proved elsewhere [M. R. Garey and D. S. Johnson, The rectilinear Steiner tree is NP-complete, SIAM J. Appl. Math., 32 (1977), pp. 826–834] and [M. R. Garey, D. S. Johnson and R. E. Tarjan, The planar Hamilton circuit problem is NP-complete, SIAM J. Comp., 5 (1976), pp. 704–714].

776 citations

••

Harry B. Hunt

^{1}, Madhav V. Marathe^{1}, Venkatesh Radhakrishnan^{1}, S. S. Ravi^{1}+2 more•Institutions (1)TL;DR: The approximation schemes for hierarchically specified unit disk graphs presented in this paper are among the first approximation schemes in the literature for natural PSPACE-hard optimization problems.

Abstract: We present NC-approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance trade-off as the best known approximation schemes for planar graphs. We also define the concept of ?-precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance trade-off than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for ?-precision unit disk graphs many more graph problems have efficient approximation schemes.Our NC-approximation schemes can also be extended to obtain efficient NC-approximation schemes for several PSPACE-hard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann, and Widmayer. The approximation schemes for hierarchically specified unit disk graphs presented in this paper are among the first approximation schemes in the literature for natural PSPACE-hard optimization problems.

332 citations