  The focus of this paper is two fold. Firstly, we present a logical approach
to graph modification problems such as minimum node deletion, edge deletion,
edge augmentation problems by expressing them as an expression in first order
(FO) logic. As a consequence, it follows that these problems have constant
factor polynomial-time approximation algorithms. In particular, node
deletion/edge deletion on a graph $G$ whose resultant is cograph, split,
threshold, comparable, interval and permutation are $O(1)$ approximable.
Secondly, we present a monadic second order (MSO) logic to minimum graph
modification problems, minimum dominating set problem and minimum coloring
problem and their variants. As a consequence, it follows that these problems
have linear-time algorithms on bounded tree-width graphs. In particular, we
show the existance of linear-time algorithms on bounded tree-width graphs for
star coloring, cd-coloring, rainbow coloring, equitable coloring, total
dominating set, connected dominating set. In a nut shell, this paper presents a
unified framework and an algorithmic scheme through logical expressions for
some graph problems through FO and MSO.
