\(R\)-Total Domination on Convex Bipartite Graphs

Let \( \mathcal{P} = \{I, I_1+d, I_1+2d, \ldots, I_1+(\ell-1)d\} \), where \( \ell, d, I_1 \) are fixed integers and \( \ell, d > 0 \). Suppose that \( G = (V, E) \) is a graph and \( R \) is a labeling function which assigns an integer \( R(v) \) to each \( v \in V \). An \emph{\( R \)-total dominating function} of \( G \) is a function \( f: V \to \mathcal{P} \) such that

\[
\sum_{u \in N_G(v)} f(u) \geq R(v)
\]

for all vertices \( v \in V \), where \( N_G(v) = \{u \mid (u, v) \in E\} \). The \emph{\( R \)-total domination problem} is to find an \( R \)-total dominating function \( f \) of \( G \) such that

\[
\sum_{v \in V} f(v)
\]

is minimized. In this paper, we present a linear-time algorithm to solve the \( R \)-total domination problem on convex bipartite graphs. Our algorithm gives a unified approach to the \( k \)-total, signed total, and minus total domination problems for convex bipartite graphs.