%0 Journal Article
%T Algorithmic Aspects of Quasi-Total Roman Domination in Graphs
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A P, Venkata Subba Reddy
%A Vikas, Mangal
%D 2022
%\ 06/01/2022
%V 7
%N 1
%P 93-104
%! Algorithmic Aspects of Quasi-Total Roman Domination in Graphs
%K Domination number
%K quasi-total Roman domination
%K complexity classes
%K graph classes
%K linear programming
%R 10.22049/cco.2021.27126.1200
%X For a simple, undirected, connected graph $G$($V,E$), a function $f : V(G) rightarrow lbrace 0, 1, 2 rbrace$ which satisfies the following conditions is called a quasi-total Roman dominating function (QTRDF) of $G$ with weight $f(V(G))=sum_{v in V(G)} f(v)$. C1). Every vertex $u in V(G)$ for which $f(u) = 0$ must be adjacent to at least one vertex $v$ with $f(v) = 2$, and C2). Every vertex $u in V(G)$ for which $f(u) = 2$ must be adjacent to at least one vertex $v$ with $f(v) geq 1$. For a graph $G$, the smallest possible weight of a QTRDF of $G$ denoted $gamma_{qtR}(G)$ is known as the textit{quasi-total Roman domination number} of $G$. The problem of determining $gamma_{qtR}(G)$ of a graph $G$ is called minimum quasi-total Roman domination problem (MQTRDP). In this paper, we show that the problem of determining whether $G$ has a QTRDF of weight at most $l$ is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. On the positive side, we show that MQTRDP for threshold graphs, chain graphs and bounded treewidth graphs is linear time solvable. Finally, an integer linear programming formulation for MQTRDP is presented.
%U http://comb-opt.azaruniv.ac.ir/article_14220_e1aeb63f7a60db0a40f1bdb4295ae39a.pdf