In this thesis, we design in-memory data structures for labeled and weights trees, so that various types of path queries or operations can be supported with efficient query time. We assume the word RAM model with word size w, which permits random accesses to w-bit memory cells. Our data structures are space-efficient and many of them are even succinct. These succinct data structures occupy spac...

Let be a connected graph with vertex set and edge . A bijection from to the is labeling of The called rainbow antimagic if for any two in path , where Rainbow coloring which has labeling. Thus, every induces G weight color connection number smallest colors all colorings denoted by In this study, we studied have an exact value joint product

A vertex-coloured graph G is said to be rainbow vertex-connected, if every two vertices of are connected by a path whose internal have distinct colours. The vertex-connection number G, denoted rvc(G), the smallest colours that needed make vertex-connected. This study focuses on deriving formulas for vertex connectivity simple ladder and roach graph.

Let P be an x-monotone polygonal path in the plane. For a path Q that approximates P let WA(Q) be the area above P and below Q, and let WB(Q) be the area above Q and below P . Given P and an integer k, we show how to compute a path Q with at most k edges that minimizes WA(Q)+WB(Q). Given P and a cost C, we show how to find a path Q with the smallest possible number of edges such that WA(Q) +WB(...

An edge-colored graph G is conflict-free connected if, between each pair of distinct vertices, there exists a path containing a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the smallest number of colors that are required in order to make G conflict-free connected. A coloring of vertices of a hypergraph H = (...

A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines, In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r ∈ T is a recti...