Algorithm complexity of neighborhood total domination and (ρ,γnt)-graphs

A neighborhood total dominating set, abbreviated for NTD-set D , is a vertex set of G such that D is a dominating set with an extra property: the subgraph induced by the open neighborhood of D has no isolated vertex. The neighborhood total domination number, denoted by γ n t ( G ) , is the minimum c...

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Bibliographic Details
Published in:Journal of combinatorial optimization Vol. 35; no. 2; pp. 424 - 435
Main Authors: Lu, Changhong, Wang, Bing, Wang, Kan
Format: Journal Article
Language:English
Published: New York Springer US 01.02.2018
Springer Nature B.V
Subjects:
Combinatorics
Convex and Discrete Geometry
Graphs
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Neighborhoods
Operations Research/Decision Theory
Optimization
Theory of Computation
Trees (mathematics)
Neighborhood total domination
Labeled graph
Algorithm complexity
05C15
ISSN:1382-6905, 1573-2886
Online Access:Get full text
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Summary:A neighborhood total dominating set, abbreviated for NTD-set D , is a vertex set of G such that D is a dominating set with an extra property: the subgraph induced by the open neighborhood of D has no isolated vertex. The neighborhood total domination number, denoted by γ n t ( G ) , is the minimum cardinality of a NTD-set in G . In this paper, we prove that NTD problem is NP-complete for bipartite graphs and split graphs. Then we give a linear-time algorithm to determine γ n t ( T ) for a given tree T . Finally, we characterize a constructive property of ( γ n t , 2 γ ) -trees and provide a constructive characterization for ( ρ , γ n t ) -graphs, where γ and ρ are domination number and packing number for the given graph, respectively.
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content type line 14
ISSN:1382-6905
1573-2886
DOI:10.1007/s10878-017-0181-6