Abstract:
For a connected graph G of order p ³ 2, a set S Í V(G) is an x-geodominating set of G if each vertex v Î 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. A connected x-geodominating set of G is an x-geodominating set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-geodominating set of G is defined as the connected x-geodomination number of G and is denoted by cgx(G). A connected x-geodominating set of cardinality cgx(G) is called a cgx-set of G. We determine bounds for it and find the same for some special classes of graphs. If p, a and b are positive integers such that 2 £ a £ b £ p - 1, then there exists a connected graph G of order p,gx(G) = a and cgx(G) = b for some vertex x in G. Also, if p, d and n are integers such that 2 £ d £ p - 2 and 1 £ n £ p, then there exists a connected graph G of order p, diameter d and cgx(G) = n for some vertex x in G. For positive integers r, d and n with r £ d £ 2r, there exists a connected graph G with rad G = r, diam G = d and cgx(G) = n for some vertex x in G.
Page(s):
101-114
DOI:
DOI not available
Published:
Journal: Journal of Prime Research in Mathematics, Volume: 5, Issue: 0, Year: 2009