The connected vertex monophonic number of a graph

Titus, P. and Iyappan, K. (2016) The connected vertex monophonic number of a graph. ACTA MATHEMATICA ACADEMIAE PAEDAGOGICAE NYÍREGYHÁZIENSIS, 32 (1). pp. 1-13. ISSN 0866-0174

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Abstract

For a connected graph G of order p ≥ 2 and a vertex x of G, a set S ⊆ V (G) is an x-monophonic set of G if each vertex v ∈ V (G) lies on an x − y monophonic path for some element y in S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G, denoted by mx(G). A connected x-monophonic set of G is an x-monophonic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-monophonic set of G is defined as the connected x-monophonic number of G and is denoted by cmx(G). We determine bounds for it and find the same for some special classes of graphs. If p, a and b are positive integers such that 2 ≤ a ≤ b ≤ p − 1, then there exists a connected graph G of order p, mx(G) = a and cmx(G) = b for some vertex x in G. Also, if p, dm and n are positive integers such that 2 ≤ dm ≤ p − 2 and 1 ≤ n ≤ p, then there exists a connected graph G of order p, monophonic diameter dm and cmx(G) = n for some vertex x in G.

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