In this paper the vertices of caterpillar tree are viewed with different approach and categorized into the sets , and , based on the distance parameters i.e., diameter and radius. The distance parameters have been presented with some set theory views. Here is the set of diametral vertices, is the set of central vertices and is the set of vertices which are neither central nor peripheral. Then, . The cardinality of these sets has some property and helps to specify the basic characters of caterpillar tree. A linear complexity algorithm is also designed to generate these sets.

ABSTRACT:Every graph has one or more diametral paths. A diametral path of a graph is a shortest path whose length is equal to the diameter of the graph. Let be a diametral vertex. There may be one or more diametral paths originating from . We want to find all the diametral paths, originating from . The total number of diametral paths reachable from a vertex is called the Diametral Reachable Index of that vertex, denoted . For any vertex , the , if there are no diametral paths reachable from , else we write , where is the total number of diametral paths reachable from vertex . An algorithm is developed to find DRI of each vertex of a graph, by modifying the DFS algorithm.

Let ( , ) be a unicyclic graph. A unicyclic graph is a connected graph that contains exactly one cycle. A dominating set of a graph G = (V, E) is a subset D of V, such that every vertex which is not in D is adjacent to at least one member of D. The domination number is the number of vertices in a smallest dominating set for G. In this paper I have presented an algorithmic approach to compute the domination number and the minimum domination set for the unicyclic graph. The algorithm has polynomial time complexity of ( ).