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Domination in discrete topology graphs
(Maltepe Üniversitesi, 2019) Jabor, Ali Ameer; Omran, Ahmed Abd Ali
In this paper we obtain a graph from the discrete topology under some conditions taken from composition of topology, study properties of that graph and the domination number of the discrete topology graph. Finally, the affection of the discrete topology graph domination parameter when a graph is modified by deleting or adding a vertex is studied in this paper. Definition 1. Let (X, ? ) be a topological space. Define the graph G? = (V, E) such that: V={u:u? ? , u?=? , X } E= {uv ? E(G? ) if u?v?= ?, u?=v and u,v ? ? }. Theorem 1. If (X, ? ) is a discrete space and X contains greater than or equal to three elements, then G? is a connected graph. Theorem 2. If (X, ? ) is a discrete space with |X| ? 3, then G? has no cut vertex. Theorem 3. If (X, ? ) is a discrete space with |X| ? 3, then ?(G? ) = 2. Theorem 4. ?(G? ? v) ? ?(G? ). Theorem 5. ?(G? ? e) = ?(G? ).