Forbidden family of Ph-magic graphs

Let G be a simple, finite, and undirected graph and H be a subgraph of G . The graph G admits an H -covering if every edge in G belongs to a subgraph isomorphic to H . A bijection f : V ( G )∪ E ( G )→1, n is a magic total labeling if for every subgraphs H ′ isomorphic to H , the sum of labels of all vertices and edges in H ′ is constant. If there exists such f , we say G is H -magic. A graph F is said to be a forbidden subgraph of H -magic graphs if F ⊆ G implies G is not an H -magic graph. A set that contains all forbidden subgraph of H -magic is called forbidden family of H -magic graphs, denoted by F ( H ) . In this paper, we consider F ( P h ) , where P h is a path of order h . We present some sufficient conditions of a graph being a member of F ( P h ) . Besides that, we show the uniqueness of a minimal tree which belongs to F ( P 3 ) and characterize P 3 -(super)magic trees.