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In this monograph, we have designed some sequential algorithm to solve some problems on interval graphs, permutation graphs and trapezoid graphs. In chapter 1, we have discussed the definitions, recognitions, applications, survey, etc. of the Interval, permutation and trapezoid graphs. In second chapter we have designed an O(n) time algorithm to solve minimum k-neighbourhood-covering problem on interval graphs. We also present efficient algorithms to find next-to-shortest path between any pair of vertices on permutation graphs and trapezoid graphs with n vertices which run in O(n^2) time in chapter 3 and chapter 5 respectively. In chapter 4, we present an O(n^2) time algorithm to find a minimum 2-tuple dominating set on permutation graphs with n vertices. Also in chapter 6, we present an algorithm to find a tree 4-spanner on trapezoid graphs in O(n) time, and in chapter 7, an O(n^2) time algorithm is presented to find a tree 3-spanner on trapezoid graphs, where n is the number ofvertices of the graph. Finally, chapter 8 contains some concluding remarks and scopes of further research on the problems that have been studied in the monograph.