Abstract
Let G = (V‚ E) be a simple connected graph with p vertices and q edges. If G1‚ G2‚ G3‚ …‚ Gn are connected edge disjoint subgraphs of G with E(G) = E(G1) ?E(G2) ?E(G3) ?… ?E(Gn)‚ then (G1‚ G2‚ G3‚ …, Gn) is said to be a decomposition of G. A decomposition (G1‚ G2‚ G3‚ …‚ Gn) of G is said to be an Arithmetic Decomposition if each Gi is connected and |E(Gi)| = a + (i -1)d, for every i = 1‚ 2‚ 3‚…‚n and a‚d ? ?. In this paper‚ we introduced a new concept Geometric Decomposition. A decomposition (Ga‚ Gar‚ Gar2‚ Gar3‚ …‚ Garn-1) of G is said to be a Geometric Decomposition (GD) if each Gari-1 is connected and |E(Gari-1)| = ari-1, for every i = 1‚ 2‚ 3‚…‚n and a‚ r ? ?. Clearly q = . If a = 1 and r = 2‚ then q = 2n - 1. Also we obtained necessary and sufficient conditions of complete tripartite graphs which are admitting a Geometric Decomposition.