Minimal connected restrained monophonic sets in graphs

• Santhakumaran, A. P. ^[1] ; Titus, P. ^[2] ; Ganesamoorthy, K. ^[3]
  1. [1] Hindustan Institute of Technology and Science.
  2. [2] University College of Engineering Nagercoil.
  3. [3] Coimbatore Institute of Technology.

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• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 41, Nº. 4, 2022, págs. 879-890
• Idioma: inglés
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• Resumen

  • For a connected graph G = (V,E) of order at least two, a connected restrained monophonic set S of G is a restrained monophonic set such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected restrained monophonic set of G is the connected restrained monophonic number of G and is denoted by mcr(G). A connected restrained monophonic set S of G is called a minimal connected restrained monophonic set if no proper subset of S is a connected restrained monophonic set of G. The upper connected restrained monophonic number of G, denoted by m+cr(G), is defined as the maximum cardinality of a minimal connected restrained monophonic set of G. We determine bounds for it and certain general properties satisfied by this parameter are studied. It is shown that, for positive integers a, b such that 4≤ a ≤ b , there exists a connected graph G such that  mcr(G) = a and m+cr(G) = b.