Dominating weakly connected set dominating bridge independent graphs

Dominating weakly connected set dominating bridge independent graphs

**Authors : **

D. Anandha Selvam, ^{1 *} and M. Davamani Christober ^{2}

**Author Address : **

^{1,2} Department of Mathematics, The American College, Madurai-625002, Tamil Nadu, India.

*Corresponding author.

**Abstract : **

A ${gamma _{wcsd}}$ set S of a connected graph $G$ is a dominating weakly connected set dominating $(wcsd)$ set of $G$ with minimum cardinality. A connected graph $G$ is a ${gamma

_{wcsd}}$ - excellent if each vertex $u$ of $G$ is in some ${gamma _{wcsd}}$ set of $G$. A graph $G$ is a ${gamma _{wcsd}}$ - flexible if to each vertex $u$ of $G$, there is a ${gamma _{wcsd}}$ set not containing $u$. A $wcsd$ set $S$ of $G$ is $wcsd$ - bridge independent set of $G$ if induced graph of $S$ contains no bridge of $G$. The minimum cardinality of a $wcsd$ -

bridge independent dominating set of $G$ is $wcsd$ - bridge independent dominating number of $G$ and is denoted $gamma _{wcsd}^{bi}(G)$. A graph $G$ is $gamma _{wcsd}^{bi}$ - excellent if every vertex $u$ of $G$ is contained in some $gamma _{wcsd}^{bi}$ - set of $G$. In this paper we have proved that (i) every graph $G$ is an induced sub graph of some $gamma

_{wcsd}^{bi}$ - excellent, ${gamma _{wcsd}}$- excellent and ${gamma _{wcsd}}$ - flexible graph H with ${gamma _{wcsd}}(G) leqslant { ext{ }}{gamma _{wcsd}}(H) leqslant ,gamma

_{wcsd}^{bi}(H) leqslant {gamma _{wcsd}}(G) + 1$ (ii) Every ${gamma _{wcsd}}$ - excellent & ${gamma _{wcsd}}$ - flexible is $gamma _{wcsd}^{bi}$ - excellent and further $gamma _{wcsd}^{bi} = {gamma _{wcsd}}$ (iii) A necessary and sufficient condition under which the graph $G = ({G_1} cup {G_2}) + uv$ where $G_1$ and $G_2$ are disjoint ${gamma _{wcsd}}$ excellent graphs and $u in V({G_1})$ & $v in V({G_2})$ is ${gamma _{wcsd}}$ excellent.

**Keywords : **

$wcsd$ - sets, ${gamma _{wcsd}}$ - sets, ${gamma _{wcsd}}$ -excellent graphs, ${gamma _{wcsd}}$ - flexible graphs, $gamma _{wcsd}^{bi}$ - sets and $gamma _{wcsd}^{bi}$ - excellent

graphs.

**DOI : **

**Article Info : **

*Received : * December 21, 2018; *Accepted : * February 11, 2019.