A Taxonomy of Perfect Domination

Abstract A variety of terminology has been used in the literature to describe a dominating set with the property that each vertex in graph is dominated exactly once (or at most once, depending on the situation). We review the various terminologies related to perfect domination, consider some of the variations of perfect domination studied in the literature, and survey some of the main results.

[1]  Jan Kratochvíl,et al.  Independent Sets with Domination Constraints , 1998, Discret. Appl. Math..

[2]  Lane H. Clark,et al.  Efficient domination of the orientations of a graph , 1998, Discret. Math..

[3]  Ghidewon Abay-Asmerom,et al.  Total Perfect Codes in Tensor Products of Graphs , 2008, Ars Comb..

[4]  Sandi Klavzar,et al.  Characterizing r-perfect codes in direct products of two and three cycles , 2005, Inf. Process. Lett..

[5]  Sandi Klavzar,et al.  An almost complete description of perfect codes in direct products of cycles , 2006, Adv. Appl. Math..

[6]  Robert R. Rubalcaba,et al.  A survey on graphs which have equal domination and closed neighborhood packing numbers , 2020 .

[7]  Jan Kratochivíl,et al.  Regular codes in regular graphs are difficult , 1994 .

[8]  Marco Cesati.  Perfect Code is W[1]-complete , 2000, Inf. Process. Lett..

[9]  Robert Cowen,et al.  Odd neighborhood transversals on grid graphs , 2007, Discret. Math..

[10]  Dieter Rautenbach,et al.  Open packing, total domination, and the P3-Radon number , 2013, Discret. Math..

[11]  Jan Arne Telle,et al.  Complexity of Domination-Type Problems in Graphs , 1994, Nord. J. Comput..

[12]  P. Cull,et al.  Perfect codes on graphs , 1997, Proceedings of IEEE International Symposium on Information Theory.

[13]  Frédéric Maffray,et al.  Exact double domination in graphs , 2005, Discuss. Math. Graph Theory.

[14]  I. Dejter,et al.  On perfect dominating sets in hypercubes and their complements , 1996 .

[15]  P. Duchet,et al.  Strongly Perfect Graphs , 1984 .

[16]  P. Delsarte.  AN ALGEBRAIC APPROACH TO THE ASSOCIATION SCHEMES OF CODING THEORY , 2011 .

[17]  C. Zheng,et al.  ; 0 ; , 1951 .

[18]  Odile Favaron,et al.  k-Domination and k-Independence in Graphs: A Survey , 2012, Graphs Comb..

[19]  Quentin F. Stout,et al.  Perfect Dominating Sets on Cube-Connected Cycles , 1993 .

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

[21]  Stephen T. Hedetniemi,et al.  Efficient Sets in Graphs , 1993, Discret. Appl. Math..

[22]  Richard C. T. Lee,et al.  The Weighted Perfect Domination Problem , 1990, Inf. Process. Lett..

[23]  R. Brualdi.  On the Diagonal Hypergraph of a Matrix , 1980 .

[24]  Frederick C. Harris,et al.  Nearly perfect sets in graphs , 1995, Discret. Math..

[25]  Kees Roos.  A note on the existence of perfect constant weight codes , 1983, Discret. Math..

[26]  Oliver Schaudt,et al.  Efficient total domination in digraphs , 2012, J. Discrete Algorithms.

[27]  C. Pandu Rangan,et al.  Weighted Independent Perfect Domination on Cocomparability Graphs , 1993, ISAAC.

[28]  Douglas F. Rall,et al.  Total Domination in Categorical Products of Graphs , 2005, Discuss. Math. Graph Theory.

[29]  Xiaoqiang Wang,et al.  Algorithms for unipolar and generalized split graphs , 2011, Discret. Appl. Math..

[30]  Wayne Goddard,et al.  Independent domination in graphs: A survey and recent results , 2013, Discret. Math..

[31]  Alice A. McRae.  Generalizing NP-completeness proofs for bipartite graphs and chordal graphs , 1995 .

[32]  S. Klavžar,et al.  1-perfect codes in Sierpiński graphs , 2002, Bulletin of the Australian Mathematical Society.

[33]  Oliver Schaudt.  On weighted efficient total domination , 2012, J. Discrete Algorithms.

[34]  Martin Knor,et al.  Efficient open domination in digraphs , 2011, Australas. J Comb..

[35]  Chuan Yi Tang,et al.  Weighted efficient domination problem on some perfect graphs , 2002, Discret. Appl. Math..

[36]  Jan Kratochvíl,et al.  Independent Sets with Domination Constraints , 2000, Discret. Appl. Math..

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

[38]  Olof Heden.  Perfect Codes in Antipodal Distance-Transitive Graphs. , 1974 .

[39]  Odile Favaron,et al.  Independent [1, k]-sets in graphs , 2014, Australas. J Comb..

[40]  Mirka Miller,et al.  Generalized Domination in Chordal Graphs , 1995, Nord. J. Comput..

[41]  Rudolf Ahlswede,et al.  On Perfect Codes and Related Concepts , 2001, Des. Codes Cryptogr..

[42]  David B. Shmoys,et al.  A Best Possible Heuristic for the k-Center Problem , 1985, Math. Oper. Res..

[43]  Quentin F. Stout,et al.  PERFECT DOMINATING SETS , 1990 .

[44]  Heather Gavlas,et al.  Efficient Open Domination , 2002, Electron. Notes Discret. Math..

[45]  Pinar Heggernes,et al.  Partitioning Graphs into Generalized Dominating Sets , 1998, Nord. J. Comput..

[46]  Gabriela Araujo-Pardo,et al.  The ωψ-perfection of graphs , 2013, Electron. Notes Discret. Math..

[47]  Maw-Shang Chang,et al.  Polynomial Algorithms for the Weighted Perfect Domination Problems on Chordal Graphs and Split Graphs , 1993, Inf. Process. Lett..