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Roman Domination of Cartesian Bundles of Cycles over Cycles

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  • Simon Brezovnik

    (Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia
    Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia)

  • Janez Žerovnik

    (Faculty of Mechanical Engineering, University of Ljubljana, 1000 Ljubljana, Slovenia
    Rudolfovo—Science and Technology Centre, 8000 Novo Mesto, Slovenia)

Abstract

A Roman dominating function f of a graph G = ( V , E ) assigns labels from the set { 0 , 1 , 2 } to vertices such that every vertex labeled 0 has a neighbor labeled 2. The weight of an RDF f is defined as w ( f ) = ∑ v ∈ V f ( v ) , and the Roman domination number, γ R ( G ) , is the minimum weight among all RDFs of G . This paper studies the domination and Roman domination numbers in Cartesian bundles of cycles. Furthermore, the constructed optimal patterns improve known bounds and suggest even better bounds might be achieved by combining patterns, especially for bundles involving shifts of order 4 k and 5 k .

Suggested Citation

  • Simon Brezovnik & Janez Žerovnik, 2025. "Roman Domination of Cartesian Bundles of Cycles over Cycles," Mathematics, MDPI, vol. 13(15), pages 1-18, July.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:15:p:2351-:d:1708086
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