Can graphs be efficiently broken down into simpler structures? This research introduces an enhanced algorithm to determine if a graph has a specific 'treewidth,' a measure of its structural complexity, within a given limit. The refined algorithm, built upon Bodlaender's method, efficiently constructs a tree decomposition if the treewidth is within the set limit, making it a significant advancement in **algorithmic graph theory**. The algorithm has an added feature: if the graph's treewidth exceeds the limit, the algorithm will return a subgraph whose treewidth exceeds the limit along with a tree decomposition of the subgraph with a width of at most twice the limit. The study's findings are especially relevant for solving the fundamental disjoint rooted paths problem, showcasing the algorithm's practical applications in solving complex problems in computer science. This enhancement allows for solving this problem in quadratic time, a considerable improvement. This research makes notable strides in the field of **graph algorithms**. Ultimately, the improved algorithm and its applications to the disjoint rooted paths problem contribute to enhanced problem solving in areas relying on efficient **tree decomposition**, including network design and computational biology. Further, it offers researchers a more robust tool for graph analysis. The advancements here have significant practical and theoretical implications for **computer science**.
Published in the International Journal of Foundations of Computer Science, this paper directly contributes to the journal's focus on theoretical computer science and algorithmic efficiency. By presenting an improved algorithm for tree decomposition, the research strengthens the journal's reputation for publishing cutting-edge advancements in graph theory and related computational problems.