[1, 2]-sets and [1, 2]-total Sets in Trees with Algorithms

A set S ? V of the graph G = ( V , E ) is called a 1 , 2 -set of G if any vertex which is not in S has at least one but no more than two neighbors in S . A set S ' ? V is called a 1 , 2 -total set of G if any vertex of G , no matter in S ' or not, is adjacent to at least one but not more than two vertices in S ' . In this paper we introduce a linear algorithm for finding the cardinality of the smallest 1 , 2 -sets and 1 , 2 -total sets of a tree and extend it to a more generalized version for i , j -sets, a generalization of 1 , 2 -sets. This answers one of the open problems proposed in Chellali et?al. (2013). Then since not all trees have 1 , 2 -total sets, we devise a recursive method for generating all the trees that do have such sets. This method also constructs every 1 , 2 -total set of each tree that it generates.

[1]  Teresa W. Haynes,et al.  [1, 2]-sets in Graphs , 2013, Discret. Appl. Math..

[2]  Michael A. Henning,et al.  RAINBOW DOMINATION IN GRAPHS , 2008 .

[3]  Gerard J. Chang,et al.  On the mixed domination problem in graphs , 2013, Theor. Comput. Sci..

[4]  Zhenbing Zeng,et al.  Hardness results and approximation algorithms for (weighted) paired-domination in graphs , 2009, Theor. Comput. Sci..

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

[6]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[7]  Ellis Horowitz,et al.  Fundamentals of Data Structures , 1984 .

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

[9]  Bhawani Sankar Panda,et al.  Complexity of distance paired-domination problem in graphs , 2012, Theor. Comput. Sci..

[10]  William Duckworth,et al.  Minimum independent dominating sets of random cubic graphs , 2002, Random Struct. Algorithms.

[11]  Ermelinda DeLaViña,et al.  On Total Domination in Graphs , 2012 .

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

[13]  Stephen T. Hedetniemi,et al.  Total domination in graphs , 1980, Networks.

[14]  Michael R. Fellows,et al.  Perfect domination , 1991, Australas. J Comb..

[15]  Yancai Zhao,et al.  The algorithmic complexity of mixed domination in graphs , 2011, Theor. Comput. Sci..

[16]  Michel Mollard,et al.  The domination number of Cartesian product of two directed paths , 2014, J. Comb. Optim..

[17]  Robert W. Irving On Approximating the Minimum Independent Dominating Set , 1991, Inf. Process. Lett..

[18]  Robert B. Allan,et al.  On domination and independent domination numbers of a graph , 1978, Discret. Math..

[19]  K. Kayathri,et al.  (1, 2)-Domination in Graphs , 2016, ICTCSDM.