# 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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Main Authors: , Article English 2009-01-01 Opuscula Mathematica http://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2934.pdf
id doaj-757cff1086b04c36aae77c7b757fb292 Article doaj-757cff1086b04c36aae77c7b757fb2922020-11-25T02:17:15ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742009-01-01294427441http://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 DOAJ English Article DOAJ A. P. Santhakumaran J. John 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 A. P. Santhakumaran J. John A. P. Santhakumaran The upper edge geodetic number and the forcing edge geodetic number of a graph The upper edge geodetic number and the forcing edge geodetic number of a graph The upper edge geodetic number and the forcing edge geodetic number of a graph The upper edge geodetic number and the forcing edge geodetic number of a graph The upper edge geodetic number and the forcing edge geodetic number of a graph upper edge geodetic number and the forcing edge geodetic number of a graph AGH Univeristy of Science and Technology Press Opuscula Mathematica 1232-9274 2009-01-01 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 $$g_1(G)=b$$. geodetic number edge geodetic basis edge geodetic number upper edge geodetic number forcing edge geodetic number http://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2934.pdf AT apsanthakumaran theupperedgegeodeticnumberandtheforcingedgegeodeticnumberofagraph AT jjohn theupperedgegeodeticnumberandtheforcingedgegeodeticnumberofagraph AT apsanthakumaran upperedgegeodeticnumberandtheforcingedgegeodeticnumberofagraph AT jjohn upperedgegeodeticnumberandtheforcingedgegeodeticnumberofagraph 1724887355387543552