Fair Domination Number in Cactus Graphs

Open access

Abstract

For k ≥ 1, a k-fair dominating set (or just kFD-set) in a graph G is a dominating set S such that |N(v) ∩ S| = k for every vertex vV \ S. The k-fair domination number of G, denoted by fdk(G), is the minimum cardinality of a kFD-set. A fair dominating set, abbreviated FD-set, is a kFD-set for some integer k ≥ 1. The fair domination number, denoted by fd(G), of G that is not the empty graph, is the minimum cardinality of an FD-set in G. In this paper, aiming to provide a particular answer to a problem posed in [Y. Caro, A. Hansberg and M.A. Henning, Fair domination in graphs, Discrete Math. 312 (2012) 2905–2914], we present a new upper bound for the fair domination number of a cactus graph, and characterize all cactus graphs G achieving equality in the upper bound of fd1(G).

[1] Y. Caro, A. Hansberg and M.A. Henning, Fair domination in graphs, Discrete Math. 312 (2012) 2905–2914. doi:10.1016/j.disc.2012.05.006

[2] B. Chaluvaraju, M. Chellali and K.A. Vidya, Perfect k-domination in graphs, Australas. J. Combin. 48 (2010) 175–184.

[3] B. Chaluvaraju and K.A. Vidya, Perfect dominating set graph of a graph G, Adv. Appl. Discrete Math. 2 (2008) 49–57.

[4] E.J. Cockayne, B.L. Hartnell, S.T. Hedetniemi and R. Laskar, Perfect domination in graphs, J. Comb. Inf. Syst. Sci. 18 (1993) 136–148.

[5] I.J. Dejter, Perfect domination in regular grid graphs, Australas. J. Combin. 42 (2008) 99–114.

[6] I.J. Dejter and A.A. Delgado, Perfect domination in rectangular grid graphs, J. Combin. Math. Combin. Comput. 70 (2009) 177–196.

[7] M.R. Fellows and M.N. Hoover, Perfect domination, Australas. J. Combin. 3 (1991) 141–150.

[8] M. Hajian and N. Jafari Rad, Trees and unicyclic graphs with large fair domination number, Util. Math. accepted.

[9] H. Hatami and P. Hatami, Perfect dominating sets in the Cartesian products of prime cycles, Electron. J. Combin. 14 (2007) #N8.

[10] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker Inc., New York, 1998).

Discussiones Mathematicae Graph Theory

The Journal of University of Zielona Góra

Journal Information


IMPACT FACTOR 2017: 0.601
5-year IMPACT FACTOR: 0.535



CiteScore 2017: 0.64

SCImago Journal Rank (SJR) 2017: 0.633
Source Normalized Impact per Paper (SNIP) 2017: 1.095

Mathematical Citation Quotient (MCQ) 2017: 0.36

Target Group

researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 3857 3857 364
PDF Downloads 62 62 16