# Distinct edge geodetic decomposition in graphs

Document Type : Original paper

Authors

1 Goverment College of Engineering, Tirunelveli

2 Bharathiyar University

Abstract

Let  $G = (V, E)$ be a simple connected graph of order $p$ and size $q$. A decomposition of a graph $G$  is a collection $\pi$ of edge-disjoint subgraphs $G_{1}, G_{2} ,\dots, G_{n}$  of $G$  such that every edge of $G$  belongs to exactly one $G_{i},(1\leq i\leq n)$. The decomposition  $\pi=\{G_{1},G_{2},\dots,G_{n}\}$ of a connected graph $G$ is said to be a distinct edge geodetic decomposition if $g_{1}(G_{i})\neq g_{1}(G_{j}),(1\leq i\neq j\leq n)$. The maximum cardinality of $\pi$ is called the distinct edge geodetic decomposition number of $G$ and is denoted by $\pi_{dg_{1}}(G)$, where $g_{1}(G)$ is the edge geodetic number of $G$. Some general properties satisfied by this concept are studied. Connected graphs of $\pi_{dg_{1}}(G)\geq2$ are characterized and connected  graphs of order $p$ with $\pi_{dg_{1}}(G)=p-2$ are characterized.

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