Independent domination in subcubic graphs

A. Akbari (), S. Akbari (), A. Doosthosseini (), Z. Hadizadeh (), Michael A. Henning () and A. Naraghi ()
Additional contact information
A. Akbari: Sharif University of Technology
S. Akbari: Sharif University of Technology
A. Doosthosseini: Sharif University of Technology
Z. Hadizadeh: Sharif University of Technology
Michael A. Henning: University of Johannesburg
A. Naraghi: Sharif University of Technology

Journal of Combinatorial Optimization, 2022, vol. 43, issue 1, No 2, 28-41

Abstract: Abstract A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. In Goddard and Henning (Discrete Math 313:839–854, 2013) conjectured that if G is a connected cubic graph of order n, then $$i(G) \le \frac{3}{8}n$$ i ( G ) ≤ 3 8 n , except if G is the complete bipartite graph $$K_{3,3}$$ K 3 , 3 or the 5-prism $$C_5 \, \Box \, K_2$$ C 5 □ K 2 . Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. In this paper, we provide a new family of connected cubic graphs G of order n such that $$i(G) = \frac{3}{8}n$$ i ( G ) = 3 8 n . We also show that if G is a subcubic graph of order n with no isolated vertex, then $$i(G) \le \frac{1}{2}n$$ i ( G ) ≤ 1 2 n , and we characterize the graphs achieving equality in this bound.

Keywords: Independent domination; Cubic graph; Subcubic graph; 05C69 (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-021-00743-z

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