IJPAM: Volume 101, No. 5 (2015)

EFFICIENT WEAK ROMAN DOMINATION IN GRAPHS

P. Roushini Leely Pushpam$^1$, M. Kamalam$^2$
$^1$Department of Mathematics
D.B. Jain College, Chennai-600 097
Tamil Nadu, INDIA
$^2$Department of Mathematics
S.S. Shasun Jain College
Chennai, 600 017, Tamil Nadu, India.


Abstract. Let $G = (V, E)$ be a graph and $f$ be a function $f : V \to \{0, 1, 2\}$. A vertex $u$ with $f(u) = 0$ is said to be undefended with respect to $f$, if it is not adjacent to a vertex with positive weight. The function $f$ is a weak Roman dominating function (WRDF) if each vertex $u$ with $f(u) = 0$ is adjacent to a vertex $v$ with $f(v) > 0$ such that the function $f^\prime : V \to \{0, 1, 2\}$ defined by $f^\prime(u) = 1$, $f^\prime(v) = f(v) - 1$ and $f^\prime(w) = f(w)$ if $w \in V - \{u, v\}$, has no undefended vertex. The efficient domination number of a graph, denoted by $F(G)$, is the maximum number of vertices that can be dominated by a set $S$ that dominates each vertex at most once. In this paper, we extend the idea of efficiency to weak Roman domination. We also characterize certain classes of graphs that are efficiently weak Roman dominatable.

Received: March 12, 2015

AMS Subject Classification: 05C69

Key Words and Phrases: domination number; weak Roman domination number; weak Roman influence

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DOI: 10.12732/ijpam.v101i5.11 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 101
Issue: 5
Pages: 701 - 710


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