A Linear-Time Algorithm for the Weighted Paired-Domination Problem on Block Graphs

TL;DR

This paper introduces a linear-time algorithm for solving the weighted paired-domination problem specifically on block graphs, improving computational efficiency over previous methods.

Contribution

The paper presents a novel $O(n+m)$-time dynamic programming algorithm for weighted paired-domination on block graphs, with an optimized $O(n)$ implementation given the block-cut-vertex structure.

Findings

Algorithm runs in linear time $O(n+m)$

Improves upon previous algorithms in efficiency

Can be completed in $O(n)$ with block-cut-vertex structure

Abstract

In a graph $G = (V,E)$, a vertex subset $S\subseteq V(G)$ is said to be a dominating set of $G$ if every vertex not in $S$ is adjacent to a vertex in $S$. A dominating set $S$ of $G$ is called a paired-dominating set of $G$ if the induced subgraph $G[S]$ contains a perfect matching. In this paper, we propose an $O(n+m)$-time algorithm for the weighted paired-domination problem on block graphs using dynamic programming, which strengthens the results in [Theoret. Comput. Sci., 410(47--49):5063--5071, 2009] and [J. Comb. Optim., 19(4):457--470, 2010]. Moreover, the algorithm can be completed in $O(n)$ time if the block-cut-vertex structure of $G$ is given.