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The Upper Connected Vertex Detour Monophonic Number of a Graph

Author

Listed:
  • S. Arumugam

    (Kalasalingam University)

  • P. Balakrishnan

    (Anna University, Tirunelveli Region)

  • A. P. Santhakumaran

    (Hindustan University, Hindustan Institute of Technology and Science)

  • P. Titus

    (Anna University, Tirunelveli Region)

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.

Suggested Citation

  • S. Arumugam & P. Balakrishnan & A. P. Santhakumaran & P. Titus, 2018. "The Upper Connected Vertex Detour Monophonic Number of a Graph," Indian Journal of Pure and Applied Mathematics, Springer, vol. 49(2), pages 365-379, June.
    • Handle: RePEc:spr:indpam:v:49:y:2018:i:2:d:10.1007_s13226-018-0274-7
    DOI: 10.1007/s13226-018-0274-7
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