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The Upper Connected Vertex Detour Monophonic Number of a Graph

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Abstract

For any vertex x in a connected graph G of order n ≥ 2, a set S x ⊆ V (G) is an x-detour monophonic set of G if each vertex v ∈ V (G) lies on an x-y detour monophonic path for some element y in S x . The minimum cardinality of an x-detour monophonic set of G is the x-detour monophonic number of G, denoted by dm x (G). A connected x-detour monophonic set of G is an x-detour monophonic set S x such that the subgraph induced by S x is connected. The minimum cardinality of a connected x-detour monophonic set of G is the connected x-detour monophonic number of G, denoted by cdm x (G). A connected x-detour monophonic set S x of G is called a minimal connected x-detour monophonic set if no proper subset of S x is a connected x-detour monophonic set. The upper connected x-detour monophonic number of G, denoted by cdm+ x (G), is defined to be the maximum cardinality of a minimal connected x-detour monophonic set of G. We determine bounds and exact values of these parameters for some special classes of graphs. We also prove that for positive integers r,d and k with 2 ≤ r ≤ d and k ≥ 2, there exists a connected graph G with monophonic radius r, monophonic diameter d and upper connected x-detour monophonic number k for some vertex x in G. Also, it is shown that for positive integers j,k,l and n with 2 ≤ j ≤ k ≤ l ≤ n - 3, there exists a connected graph G of order n with dm x (G) = j,dm+ x (G) = k and cdm+ x (G) = l for some vertex x in G.

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Authors and Affiliations

  1. National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Anand Nagar, Krishnankoil, 626 126, Tamil Nadu, India

    S. Arumugam

  2. Department of Mathematics, University College of Engineering Nagercoil, Anna University, Tirunelveli Region, Negercoil, 629 004, India

    P. Balakrishnan & P. Titus

  3. Department of Mathematics, Hindustan University, Hindustan Institute of Technology and Science, Chennai, 603 103, India

    A. P. Santhakumaran

Authors
  1. S. Arumugam
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  2. P. Balakrishnan
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  3. A. P. Santhakumaran
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  4. P. Titus
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Correspondence to S. Arumugam.

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Also at Department of Computer Science, Liverpool Hope University, Liverpool, UK; Department of Computer Science, Ball State University, USA.

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Arumugam, S., Balakrishnan, P., Santhakumaran, A.P. et al. The Upper Connected Vertex Detour Monophonic Number of a Graph. Indian J Pure Appl Math 49, 365–379 (2018). https://doi.org/10.1007/s13226-018-0274-7

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  • DOI: https://doi.org/10.1007/s13226-018-0274-7

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