TY - JOUR
ID - 26066
TI - Independent roman $\{3\}$-domination
JO - Transactions on Combinatorics
JA - TOC
LA - en
SN - 2251-8657
AU - Chakradhar, P.
AU - Venkata Subba Reddy, P.
AD - Department of Computer Science and Engineering, SR University, Warangal - 506 371, India
AD - Department of Computer Science and Engineering, National Institute of Technology, Warangal - 506 004, India
Y1 - 2022
PY - 2022
VL - 11
IS - 2
SP - 99
EP - 110
KW - Roman {3}-domination
KW - Independent Roman {3}-domination
KW - NP-complete
KW - APX-hard
KW - Integer Linear Programming
DO - 10.22108/toc.2021.127953.1840
N2 - Let $G$ be a simple, undirected graph. In this paper, we initiate the study of independent Roman $\{3\}$-domination. A function $g : V(G) \rightarrow \lbrace 0, 1, 2, 3 \rbrace$ having the property that $\sum_{v \in N_G(u)}^{} g(v) \geq 3$, if $g(u) = 0$, and $\sum_{v \in N_G(u)}^{} g(v) \geq 2$, if $g(u) = 1$ for any vertex $u \in V(G)$, where $N_G(u)$ is the set of vertices adjacent to $u$ in $G$, and no two vertices assigned positive values are adjacent is called an \textit{ independent Roman $\{3\}$-dominating function} (IR3DF) of $G$. The weight of an IR3DF $g$ is the sum $g(V) = \sum_{v \in V}g(v)$. Given a graph $G$ and a positive integer $k$, the independent Roman $\{3\}$-domination problem (IR3DP) is to check whether $G$ has an IR3DF of weight at most $k$. We investigate the complexity of IR3DP in bipartite and chordal graphs. The minimum independent Roman $\lbrace 3 \rbrace$-domination problem (MIR3DP) is to find an IR3DF of minimum weight in the input graph. We show that MIR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs. We also show that the domination problem and IR3DP are not equivalent in computational complexity aspects. Finally, we present an integer linear programming formulation for MIR3DP.
UR - https://toc.ui.ac.ir/article_26066.html
L1 - https://toc.ui.ac.ir/article_26066_8685d9cd7d13606daa8d949bab0fa40a.pdf
ER -