Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21286220211201Distinct edge geodetic decomposition in graphs1851961411410.22049/cco.2020.26638.1126ENJ. JOHNGoverment College of Engineering, Tirunelveli0000-0001-5528-4387D. StalinBharathiyar UniversityJournal Article20190811Let $G = (V, E)$ 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.https://comb-opt.azaruniv.ac.ir/article_14114_9555441344465ec52e1ff6583ab21566.pdf