Revisiting path-type covering and partitioning problems

TL;DR

This paper reviews recent developments in various path cover and partitioning problems in graph theory, aiming to unify and classify the expanding literature on these topics and their inter-relationships.

Contribution

It provides a comprehensive summary, classification, and correlation of recent research on path cover and partitioning problems in graph theory.

Findings

Summarizes recent advances in path cover and partitioning problems.

Classifies different types of path-related covering and partitioning problems.

Analyzes inter-relationships among various graph covering concepts.

Abstract

Covering problems belong to the foundation of graph theory. There are several types of covering problems in graph theory such as covering the vertex set by stars (domination problem), covering the vertex set by cliques (clique covering problem), covering the vertex set by independent sets (coloring problem), and covering the vertex set by paths or cycles. A similar concept which is partitioning problem is also equally important. Lately research in graph theory has produced unprecedented growth because of its various application in engineering and science. The covering and partitioning problem by paths itself have produced a sizable volume of literatures. The research on these problems is expanding in multiple directions and the volume of research papers is exploding. It is the time to simplify and unify the literature on different types of the covering and partitioning problems. The problems considered in this article are path cover problem, induced path cover problem, isometric path cover problem, path partition problem, induced path partition problem and isometric path partition problem. The objective of this article is to summarize the recent developments on these problems, classify their literatures and correlate the inter-relationship among the related concepts.