# The monophonie detour hull number of a graph

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Amrita University,

Amrita Vishwa Vidyapeetham

Amritanagar, Coimbatore - 641 112

Tamilnadu, India

Amrita Vishwa Vidyapeetham

Amritanagar, Coimbatore - 641 112

Tamilnadu, India

## Publication Type:

Journal Article## Authors:

Santhakumaran, A.P.a; Jebaraj, T.b; Chandran, S.V.U.c## Source:

Journal of Combinatorial Mathematics and Combinatorial Computing, Volume 84, p.155-165 (2013)## URL:

http://www.scopus.com/inward/record.url?eid=2-s2.0-84874922640&partnerID=40&md5=8f399966aa3df261242fde637096e620## Keywords:

Convex set, Graph theory, Hull number, M-detour, M-detour extreme vertex, M-detour number, Set theory## Abstract:

{For vertices u and v in a connected graph G = (V, E), the monophonic detour distance dm(u,v) is the length of a longest u-v monophonic path in G. An u-v monophonie path of length dm (u,v) is an u-v monophonie detour or an u-v m-detour. The set /dm[u,n] consists of all those vertices lying on an u-v m-detour in G. Given a set S of vertices of G, the union of all sets Idm [u, v] for u, v 6 S, is denoted by Idm [S]. A set S is an m-detour convex set if Idm [S] = S. The m-detour convex hull [5]dm of 5 in G is the smallest m-detour convex set containing S. A set S of vertices of G is an m-detour set if Idm [S] = V and the minimum cardinality of an m-detour set is the m-detour number md(G) of G. A set 5 of vertices of G is an m-detour hull set if [S]dm = V and the minmimum cardinality of an m-detour hull set is the m-detour hull number md h(G) of G. Certain general properties of these concepts are studied. Bounds for the m-detour hull number of a graph are obtained. It is proved that every two integers a and b with 2 ≤ a ≤ b are realizable as the m-detour hull number and the m-detour number respectively, of some graph. Graphs G of order n for which mdj,(G) = n or mdh(G) = n - 1 are characterized. It is proved that for each triple a, b and k of positive integers with a < b and k ≥ 3, there exists a connected graph G with radmG = a

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## Cite this Research Publication

A. Pa Santhakumaran, Jebaraj, Tb, and Chandran, S. V. Uc, “The monophonie detour hull number of a graph”, Journal of Combinatorial Mathematics and Combinatorial Computing, vol. 84, pp. 155-165, 2013.## Related Research Publications