Abstract:
This paper presents significant contributions to the study of the Minimum Eccentricity Shortest Path (MESP) problem in graph theory, with a focus on polynomial kernelization, structural parameters, and special graph classes. The introductory chapters provide a foundation in graph theory, parameterized complexity, and W-hardness, laying the groundwork for a thorough exploration of algorithmic complexities and catches us up to the latest known results on Minimum Eccentricity Shortest Path. In Chapter 3, we introduce a polynomial kernel parameterized by neighborhood diversity, grounded in two foundational claims. One establishes a crucial restriction on the intersection between a module and an MESP, limiting it to at most one vertex and the other extends this understanding, asserting that if a module contains a vertex on an MESP, all other module vertices lie on an alternative MESP with the same eccentricity. Building upon these claims, we propose a reduction rule that forms the basis of a polynomial kernel for the MESP problem. In the same chapter, we present an algorithm for finding MESP with eccentricity 1 in Proper Interval Graphs, offering insights into consecutive connecting cliques representations. The algorithm’s correctness is rigorously proven, and an extension algorithm for general eccentricities is introduced. The efficiency of these algorithms is demonstrated through time complexity analysis. In chapter 4, conclusion, our work contributes to the theoretical understanding of graph structures and polynomial kernelization. The proposed reduction rule streamlines the analysis of specific MESP instances, and the algorithms offer practical solutions for Proper Interval Graphs. Future research directions include exploring MESP in diverse graph classes, developing fpt algorithms, investigating polynomial kernelization for distance to proper interval graphs, and applying these findings to real-world scenarios.
