IJPAM: Volume 106, No. 2 (2016)

$1$-MOVABLE CLIQUE DOMINATING SETS OF A GRAPH

Teffany V. Daniel$^1$, Sergio R. Canoy, Jr.$^2$
$^1$Department of Mathematics
Bukidnon State University
Malaybalay City, PHILIPPINES
$^2$Department of Mathematics and Statistics
Mindanao State University-Iligan Institute of Technology
Andres Bonifacio Avenue, Tibanga, Iligan City 9200, PHILIPPINES


Abstract. A clique (convex) dominating set $S$ of $G$ is a $1$-movable clique dominating set (resp. $1$-movable convex dominating set) of $G$ if for every $v \in S$, either $S\setminus \{v\}$ is a clique (resp. convex) dominating set or there exists a vertex $u\in (V(G)\setminus S)\cap N_G(v)$ such that $(S\setminus \{v\})\cup \{u\}$ is a clique (resp. convex) dominating set of $G$. The minimum cardinality of a $1$-movable clique (resp. $1$-movable convex) dominating set of $G$, denoted by $\gamma_{mcl}^1(G)$ (resp. $\gamma_{mcon}^1(G)$), is called the $1$-movable clique domination number (resp. $1$-movable convex domination number) of $G$. A $1$-movable clique dominating set in $G$ with cardinality $\gamma_{mcl}^1(G)$ is called a $\gamma_{mcl}^1$-set of $G$.

This paper aims to characterize the $1$-movable clique dominating sets of some graphs including those resulting from the join and composition of two graphs. The corresponding $1$-movable clique domination number of the resulting graph is then determined. Further, it is shown that the concepts of $1$-movable clique domination and $1$-movable convex domination are equivalent.

Received: September 2, 2015

AMS Subject Classification: 05C69

Key Words and Phrases: clique domination, convex domination, $1$-movable clique domination, $1$-movable convex domination

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DOI: 10.12732/ijpam.v106i2.10 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: 2016
Volume: 106
Issue: 2
Pages: 463 - 471


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