%0 Journal Article
%T THE UPPER OPEN GEODETIC NUMBER OF A GRAPH
%A A. P. Santhakumaran
%A T. Kumari Latha
%J Zbornik Radova Ekonomskog Fakulteta u Isto£¿nom Sarajevu
%D 2012
%I Ekonomski fakultet u Isto?nom Sarajevu
%X For a connected graph G of ordern, a set S of vertices of G is a geodetic set of G ifeach vertex n of G lies on a x-y geodesic for someelements x and y in S. The minimum cardinality of ageodetic set of G is defined as the geodetic numberof G, denoted by g(G). A geodetic set of cardinalityg(G) is called a g-set of G. A set S of vertices of aconnected graph G is an open geodetic set of G iffor each vertex n in G, either n is an extreme vertexof G and n   S; or n is an internal vertex of an x-ygeodesic for some x,y S. An open geodetic set ofminimum cardinality is a minimum open geodeticset and this cardinality is the open geodeticnumber, og(G). An open geodetic set S in aconnected graph G is called a minimal opengeodetic set if no proper subset of S is an opengeodetic set of G. The upper open geodetic numberog+(G) of G is the maximum cardinality of aminimal open geodetic set of G. It is shown that, fora connected graph G of order n, og(G)=n, if andonly if og+(G)=n, and also that og(G)=3 if any onlyif og+(G)=3. It is shown that for positive integers aand b with 4 ¡Ü a ¡Ü b, there exists a connected graphG with og(G) =a and og+(G)=b. Also, it is shownthat for positive integers a, b, c with 4 ¡Ü a ¡Ü b ¡Ü cand b ¡Ü 3a, there exists a connected graph G withg(G)=a, og(G)=b and og+(G)= c.
%K geodesic
%K geodetic number
%K open geodetic number
%K upper open geodetic number.
%U http://www.ekofis.org/images/dokumenti/Aktivnosti/zr2012/31-43%20-%20A%20P%20Shantakumaran,%20T%20Kumari%20Latha.pdf