The total detour monophonic number of a graph

Abstract: For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A longest x − y monophonic path is called an x − y detour monophonic path. A s...

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Autores principales:	Santhakumaran,A. P., Titus,P., Ganesamoorthy,K.
Lenguaje:	English
Publicado:	Universidad Católica del Norte, Departamento de Matemáticas 2017
Materias:	Detour monophonic set detour monophonic number total detour monophonic set total detour monophonic number
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200209
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spelling	oai:scielo:S0716-091720170002002092017-07-06The total detour monophonic number of a graphSanthakumaran,A. P.Titus,P.Ganesamoorthy,K. Detour monophonic set detour monophonic number total detour monophonic set total detour monophonic number Abstract: For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A longest x − y monophonic path is called an x − y detour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G lies on an x − y detour monophonic path for some x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G). A total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dm t (G). A total detour monophonic set of cardinality dm t (G) is called a dm t -set of G. We determine bounds for it and characterize graphs which realize the lower bound. It is shown that for positive integers r, d and k ≥ 6 with r < d there exists a connected graph G with monophonic radius r, monophonic diameter d and dm t (G) = k. For positive integers a, b such that 4 ≤ a ≤ b with b ≤ 2a, there exists a connected graph G such that dm(G) = a and dm t (G) = b. Also, if p, d and k are positive integers such that 2 ≤ d ≤ p − 2, 3 ≤ k ≤ p and p − d − k + 3 ≥ 0, there exists a connected graph G of order p, monophonic diameter d and dm t (G) = k.info:eu-repo/semantics/openAccessUniversidad Católica del Norte, Departamento de MatemáticasProyecciones (Antofagasta) v.36 n.2 20172017-06-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200209en10.4067/S0716-09172017000200209
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Detour monophonic set detour monophonic number total detour monophonic set total detour monophonic number
spellingShingle	Detour monophonic set detour monophonic number total detour monophonic set total detour monophonic number Santhakumaran,A. P. Titus,P. Ganesamoorthy,K. The total detour monophonic number of a graph
description	Abstract: For a connected graph G = (V, E) of order at least two, a chord of a path P is an edge joining two non-adjacent vertices of P. A path P is called a monophonic path if it is a chordless path. A longest x − y monophonic path is called an x − y detour monophonic path. A set S of vertices of G is a detour monophonic set of G if each vertex v of G lies on an x − y detour monophonic path for some x and y in S. The minimum cardinality of a detour monophonic set of G is the detour monophonic number of G and is denoted by dm(G). A total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dm t (G). A total detour monophonic set of cardinality dm t (G) is called a dm t -set of G. We determine bounds for it and characterize graphs which realize the lower bound. It is shown that for positive integers r, d and k ≥ 6 with r < d there exists a connected graph G with monophonic radius r, monophonic diameter d and dm t (G) = k. For positive integers a, b such that 4 ≤ a ≤ b with b ≤ 2a, there exists a connected graph G such that dm(G) = a and dm t (G) = b. Also, if p, d and k are positive integers such that 2 ≤ d ≤ p − 2, 3 ≤ k ≤ p and p − d − k + 3 ≥ 0, there exists a connected graph G of order p, monophonic diameter d and dm t (G) = k.
author	Santhakumaran,A. P. Titus,P. Ganesamoorthy,K.
author_facet	Santhakumaran,A. P. Titus,P. Ganesamoorthy,K.
author_sort	Santhakumaran,A. P.
title	The total detour monophonic number of a graph
title_short	The total detour monophonic number of a graph
title_full	The total detour monophonic number of a graph
title_fullStr	The total detour monophonic number of a graph
title_full_unstemmed	The total detour monophonic number of a graph
title_sort	total detour monophonic number of a graph
publisher	Universidad Católica del Norte, Departamento de Matemáticas
publishDate	2017
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172017000200209
work_keys_str_mv	AT santhakumaranap thetotaldetourmonophonicnumberofagraph AT titusp thetotaldetourmonophonicnumberofagraph AT ganesamoorthyk thetotaldetourmonophonicnumberofagraph AT santhakumaranap totaldetourmonophonicnumberofagraph AT titusp totaldetourmonophonicnumberofagraph AT ganesamoorthyk totaldetourmonophonicnumberofagraph
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The total detour monophonic number of a graph

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