The upper edge geodetic number and the forcing edge geodetic number of a graph
An edge geodetic set of a connected graph \(G\) of order \(p \geq 2\) is a set \(S \subseteq V(G)\) such that every edge of \(G\) is contained in a geodesic joining some pair of vertices in \(S\). The edge geodetic number \(g_1(G)\) of \(G\) is the minimum cardinality of its edge geodetic sets and a...
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AGH Univeristy of Science and Technology Press
20090101

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doaj757cff1086b04c36aae77c7b757fb29220201125T02:17:15ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232927420090101294427441http://dx.doi.org/10.7494/OpMath.2009.29.4.4272934The upper edge geodetic number and the forcing edge geodetic number of a graphA. P. Santhakumaran0J. John1St. Xavier’s College (Autonomous), Research Department of Mathematics, Palayamkottai  627 002, IndiaAlagappa Chettiar Govt. College of Engineering & Technology, Department of Mathematics, Karaikudi  630 004, IndiaAn edge geodetic set of a connected graph \(G\) of order \(p \geq 2\) is a set \(S \subseteq V(G)\) such that every edge of \(G\) is contained in a geodesic joining some pair of vertices in \(S\). The edge geodetic number \(g_1(G)\) of \(G\) is the minimum cardinality of its edge geodetic sets and any edge geodetic set of cardinality \(g_1(G)\) is a minimum edge geodetic set of \(G\) or an edge geodetic basis of \(G\). An edge geodetic set \(S\) in a connected graph \(G\) is a minimal edge geodetic set if no proper subset of \(S\) is an edge geodetic set of \(G\). The upper edge geodetic number \(g_1^+(G)\) of \(G\) is the maximum cardinality of a minimal edge geodetic set of \(G\). The upper edge geodetic number of certain classes of graphs are determined. It is shown that for every two integers \(a\) and \(b\) such that \(2 \leq a \leq b\), there exists a connected graph \(G\) with \(g_1(G)=a\) and \(g_1^+(G)=b\). For an edge geodetic basis \(S\) of \(G\), a subset \(T \subseteq S\) is called a forcing subset for \(S\) if \(S\) is the unique edge geodetic basis containing \(T\). A forcing subset for \(S\) of minimum cardinality is a minimum forcing subset of \(S\). The forcing edge geodetic number of \(S\), denoted by \(f_1(S)\), is the cardinality of a minimum forcing subset of \(S\). The forcing edge geodetic number of \(G\), denoted by \(f_1(G)\), is \(f_1(G) = min\{f_1(S)\}\), where the minimum is taken over all edge geodetic bases \(S\) in \(G\). Some general properties satisfied by this concept are studied. The forcing edge geodetic number of certain classes of graphs are determined. It is shown that for every pair \(a\), \(b\) of integers with \(0 \leq a \lt b\) and \(b \geq 2\), there exists a connected graph \(G\) such that \(f_1(G)=a\) and \(g_1(G)=b\).http://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2934.pdfgeodetic numberedge geodetic basisedge geodetic numberupper edge geodetic numberforcing edge geodetic number 
collection 
DOAJ 
language 
English 
format 
Article 
sources 
DOAJ 
author 
A. P. Santhakumaran J. John 
spellingShingle 
A. P. Santhakumaran J. John The upper edge geodetic number and the forcing edge geodetic number of a graph Opuscula Mathematica geodetic number edge geodetic basis edge geodetic number upper edge geodetic number forcing edge geodetic number 
author_facet 
A. P. Santhakumaran J. John 
author_sort 
A. P. Santhakumaran 
title 
The upper edge geodetic number and the forcing edge geodetic number of a graph 
title_short 
The upper edge geodetic number and the forcing edge geodetic number of a graph 
title_full 
The upper edge geodetic number and the forcing edge geodetic number of a graph 
title_fullStr 
The upper edge geodetic number and the forcing edge geodetic number of a graph 
title_full_unstemmed 
The upper edge geodetic number and the forcing edge geodetic number of a graph 
title_sort 
upper edge geodetic number and the forcing edge geodetic number of a graph 
publisher 
AGH Univeristy of Science and Technology Press 
series 
Opuscula Mathematica 
issn 
12329274 
publishDate 
20090101 
description 
An edge geodetic set of a connected graph \(G\) of order \(p \geq 2\) is a set \(S \subseteq V(G)\) such that every edge of \(G\) is contained in a geodesic joining some pair of vertices in \(S\). The edge geodetic number \(g_1(G)\) of \(G\) is the minimum cardinality of its edge geodetic sets and any edge geodetic set of cardinality \(g_1(G)\) is a minimum edge geodetic set of \(G\) or an edge geodetic basis of \(G\). An edge geodetic set \(S\) in a connected graph \(G\) is a minimal edge geodetic set if no proper subset of \(S\) is an edge geodetic set of \(G\). The upper edge geodetic number \(g_1^+(G)\) of \(G\) is the maximum cardinality of a minimal edge geodetic set of \(G\). The upper edge geodetic number of certain classes of graphs are determined. It is shown that for every two integers \(a\) and \(b\) such that \(2 \leq a \leq b\), there exists a connected graph \(G\) with \(g_1(G)=a\) and \(g_1^+(G)=b\). For an edge geodetic basis \(S\) of \(G\), a subset \(T \subseteq S\) is called a forcing subset for \(S\) if \(S\) is the unique edge geodetic basis containing \(T\). A forcing subset for \(S\) of minimum cardinality is a minimum forcing subset of \(S\). The forcing edge geodetic number of \(S\), denoted by \(f_1(S)\), is the cardinality of a minimum forcing subset of \(S\). The forcing edge geodetic number of \(G\), denoted by \(f_1(G)\), is \(f_1(G) = min\{f_1(S)\}\), where the minimum is taken over all edge geodetic bases \(S\) in \(G\). Some general properties satisfied by this concept are studied. The forcing edge geodetic number of certain classes of graphs are determined. It is shown that for every pair \(a\), \(b\) of integers with \(0 \leq a \lt b\) and \(b \geq 2\), there exists a connected graph \(G\) such that \(f_1(G)=a\) and \(g_1(G)=b\). 
topic 
geodetic number edge geodetic basis edge geodetic number upper edge geodetic number forcing edge geodetic number 
url 
http://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2934.pdf 
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