Connected power domination in graphs

The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from which every vertex in the graph can be observed, following a set of rules for power system monitoring. In this paper, we study the problem of finding a minimum power dominating set which is connected; the cardinality of such a set is called the connected power domination number of the graph. We show that the connected power domination number of a graph is NP-hard to compute in general, but can be computed in linear time in cactus graphs and block graphs. We also give various structural results about connected power domination, including a cut vertex decomposition and a characterization of the effects of various vertex and edge operations on the connected power domination number. Finally, we present novel integer programming formulations for power domination, connected power domination, and power propagation time, and give computational results.

[1]  Daniela Ferrero,et al.  Note on power propagation time and lower bounds for the power domination number , 2017, J. Comb. Optim..

[2]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[3]  Darren D. Row.  Zero forcing number, path cover number, and maximum nullity of cacti , 2011 .

[4]  Ashkan Aazami,et al.  Approximation Algorithms and Hardness for Domination with Propagation , 2007, SIAM J. Discret. Math..

[5]  Rolf Niedermeier,et al.  Improved Algorithms and Complexity Results for Power Domination in Graphs , 2005, FCT.

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

[7]  Gilbert Laporte,et al.  Improvements and extensions to the Miller-Tucker-Zemlin subtour elimination constraints , 1991, Oper. Res. Lett..

[8]  Michael A. Henning,et al.  Bounds on the connected domination number of a graph , 2013, Discret. Appl. Math..

[9]  Bistra N. Dilkina,et al.  Solving Connected Subgraph Problems in Wildlife Conservation , 2010, CPAIOR.

[10]  Jean Cardinal,et al.  The Price of Connectivity for Vertex Cover , 2014, Discret. Math. Theor. Comput. Sci..

[11]  Lei Huang,et al.  Optimal PMU Placement Considering Controlled Islanding of Power System , 2014, IEEE Transactions on Power Systems.

[12]  Ashkan Aazami,et al.  Hardness results and approximation algorithms for some problems on graphs , 2008 .

[13]  Wei Wang,et al.  Complexity and algorithms for the connected vertex cover problem in 4-regular graphs , 2017, Appl. Math. Comput..

[14]  Stefan Richter,et al.  Parameterized power domination complexity , 2006, Inf. Process. Lett..

[15]  André Raspaud,et al.  Generalized power domination of graphs , 2012, Discret. Appl. Math..

[16]  P. S. Georgilakis,et al.  Taxonomy of PMU Placement Methodologies , 2012, IEEE Transactions on Power Systems.

[17]  Raphael Yuster,et al.  Connected Domination and Spanning Trees with Many Leaves , 2000, SIAM J. Discret. Math..

[18]  Boris Brimkov,et al.  Restricted power domination and zero forcing problems , 2019, J. Comb. Optim..

[19]  Shaun M. Fallat,et al.  Computation of minimal rank and path cover number for certain graphs , 2004 .

[20]  David A. Bader,et al.  Benchmarking for Graph Clustering and Partitioning , 2014, Encyclopedia of Social Network Analysis and Mining.

[21]  T. Baldwin,et al.  Power system observability with minimal phasor measurement placement , 1993 .

[22]  D. T. Lee,et al.  Power Domination Problem in Graphs , 2005, COCOON.

[23]  Robert E. Tarjan,et al.  A Note on Finding the Bridges of a Graph , 1974, Inf. Process. Lett..

[24]  Vittorio Giovannetti,et al.  Full control by locally induced relaxation. , 2007, Physical review letters.

[25]  Uppaluri S. R. Murty,et al.  Graph Theory with Applications , 1978 .

[26]  Franz Franchetti,et al.  An Information-Theoretic Approach to PMU Placement in Electric Power Systems , 2012, IEEE Transactions on Smart Grid.

[27]  Hong-Gwa Yeh,et al.  On minimum rank and zero forcing sets of a graph , 2010 .

[28]  Neng Fan,et al.  Solving the Connected Dominating Set Problem and Power Dominating Set Problem by Integer Programming , 2012, COCOA.

[29]  Daniela Ferrero,et al.  Power domination and zero forcing , 2015 .

[30]  Yuanzhan Sun,et al.  Optimal PMU placement for full network observability using Tabu search algorithm , 2006 .

[31]  Paul Dorbec,et al.  Generalized Power Domination: Propagation Radius and Sierpiński Graphs , 2014 .

[32]  Chung-Shou Liao.  Power domination with bounded time constraints , 2016, J. Comb. Optim..

[33]  F Aminifar,et al.  Probabilistic Multistage PMU Placement in Electric Power Systems , 2011, IEEE Transactions on Power Delivery.

[34]  Gerard J. Chang,et al.  On the k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{k}$$\end{document}-power domination of hypergraphs , 2013, Journal of Combinatorial Optimization.

[35]  Ashkan Aazami,et al.  Domination in graphs with bounded propagation: algorithms, formulations and hardness results , 2008, J. Comb. Optim..

[36]  R. A. Zemlin,et al.  Integer Programming Formulation of Traveling Salesman Problems , 1960, JACM.

[37]  R. Kipp Martin,et al.  Using separation algorithms to generate mixed integer model reformulations , 1991, Oper. Res. Lett..

[38]  P. Nylen,et al.  Minimum-rank matrices with prescribed graph , 1996 .

[39]  Boris Brimkov,et al.  Computational Approaches for Zero Forcing and Related Problems , 2019, Eur. J. Oper. Res..

[40]  Fabrizio Grandoni,et al.  Solving Connected Dominating Set Faster than 2n , 2007, Algorithmica.

[41]  Michael A. Henning,et al.  Domination in Graphs Applied to Electric Power Networks , 2002, SIAM J. Discret. Math..

[42]  Boris Brimkov,et al.  Complexity and computation of connected zero forcing , 2017, Discret. Appl. Math..

[43]  Boting Yang.  Fast-mixed searching and related problems on graphs , 2013, Theor. Comput. Sci..

[44]  Lenwood S. Heath,et al.  The PMU Placement Problem , 2005, SIAM J. Discret. Math..

[45]  Shaun M. Fallat,et al.  On the difference between the maximum multiplicity and path cover number for tree-like graphs , 2005 .

[46]  Boris Brimkov,et al.  Graphs with Extremal Connected Forcing Numbers , 2017, ArXiv.

[47]  Leslie Hogben,et al.  Vertex and edge spread of zero forcing number, maximum nullity, and minimum rank of a graph , 2012 .

[48]  W. Haemers.  Zero forcing sets and minimum rank of graphs , 2008 .

[49]  S. C. Srivastava,et al.  Multi-criteria decision-making approach for multistage optimal placement of phasor measurement units , 2011 .

[50]  Ning Zhou,et al.  PMU placement for state estimation considering measurement redundancy and controlled islanding , 2016, 2016 IEEE Power and Energy Society General Meeting (PESGM).

[51]  Dennis J. Brueni.  Minimal PMU placement for graph observability: a decomposition approach , 1993 .

[52]  Alexandre Salles da Cunha,et al.  The k-Cardinality Tree Problem: Reformulations and Lagrangian Relaxation , 2010, Discret. Appl. Math..