  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.
