The edge fixed geodomination number of a graph
A. P. Santhakumaran, P. Titus · 2009
For a vertex x in a connected graph G = (V (G), E(G)) of order p � 3, a set SV (G) is an x-geodominating set of G if each ver- tex v 2 V (G) lies on an x-y geodesic for some element y in S. The minimum cardinality of an x-geodominating set of G is defined as the x-geodomination number of G, denoted by gx(G). An x-geodominating set of cardinality gx(G) is called a gx-set of G. For an edge e = xy in G, a set SV (G) is an e-geodominating set of G if each vertex v 2 V (G) lies on either an x z geodesic or an y z geodesic for some element z in S. The minimum cardinality of an e-geodominating set of G is defined as the e-geodomination number of G, denoted by ge(G). An e-geodominating set of cardinality ge(G) is called a ge-set of G. Some general properties satisfied by e-geodominating sets are studied. We de- termine bounds for the e-geodomination number and find the same for some special classes of graphs. For positive integers r, d and n � 2 with r < d � 2r, there exists a connected graph G with rad G = r, diam G = d and gxy(G) = n or n 1 for any edge xy in G. If p, d and n are integers such that 3 � dp 1,2 � np 2 and p d n + 1 � 0, then there exists a graph G of order p, diameter d and gxy(G) = n or n 1 for any edge xy in G.