On the Structure of Dominating Graphs

The k -dominating graph D k ( G ) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k , two such sets being adjacent if they differ by either adding or deleting a single vertex. A graph is a dominating graph if it is isomorphic to D k ( G ) for som...

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Bibliographic Details
Published inGraphs and combinatorics Vol. 33; no. 4; pp. 665 - 672
Main Authors Alikhani, Saeid, Fatehi, Davood, Klavžar, Sandi
Format Journal Article
LanguageEnglish
Published Tokyo Springer Japan 01.07.2017
Subjects
Combinatorics
Engineering Design
Mathematics
Mathematics and Statistics
Original Paper
Domination
Dominating graph
Domination polynomial
Paths and cycles
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Summary:The k -dominating graph D k ( G ) of a graph G is defined on the vertex set consisting of dominating sets of G with cardinality at most k , two such sets being adjacent if they differ by either adding or deleting a single vertex. A graph is a dominating graph if it is isomorphic to D k ( G ) for some graph G and some positive integer k . Answering a question of Haas and Seyffarth for graphs without isolates, it is proved that if G is such a graph of order n ≥ 2 and with G ≅ D k ( G ) , then k = 2 and G ≅ K 1 , n - 1 for some n ≥ 4 . It is also proved that for a given r there exist only a finite number of r -regular, connected dominating graphs of connected graphs. In particular, C 6 and C 8 are the only dominating graphs in the class of cycles. Some results on the order of dominating graphs are also obtained.
ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-017-1792-5