Journal of Prime Research in Mathematics

# The connected vertex geodomination number of a graph

A. P. Santhakumaran
1P. G. and Research Department of Mathematics, St.Xavier’s College (Autonomous),
Palayamkottai, India.
P. Titus
Department of Mathematics, Anna University Tirunelveli, Tirunelveli, India.

$$^{1}$$Corresponding Author: apskumar1953@yahoo.co.in

### Abstract

For a connected graph $$G$$ of order $$p ≥ 2$$, a set $$S ⊆ V (G)$$ is an $$x$$-geodominating set of $$G$$ if each vertex $$v ∈ V (G)$$ 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 $$g_x(G)$$ is called a $$g_x$$-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 $$cg_x(G)$$. A connected $$x$$-geodominating set of cardinality $$cg_x(G)$$ is called a $$cg_x$$-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 $$p$$, $$g_x(G) = a$$ and $$cg_x(G) = b$$ 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 $$cg_x(G) = n$$ 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 $$cg_x(G) = n$$ for some vertex $$x$$ in $$G$$.

#### Keywords:

Geodesic, vertex geodomination number, connected vertex geodomination number.