Forbidden family of Ph-magic graphs

Maryati, Tita Khalis (Department of Mathematics Education, Universitas Islam Negeri Syarif Hidayatullah Jakarta, Indonesia)
Hadiputra, Fawwaz Fakhrurrozi (Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia)
Salman, A. N. M. (Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Indonesia)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 12, No 1 (2024): Electronic Journal of Graph Theory and Applications

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(Ph), where Ph is a path of order h. We present some sufficient conditions of a graph being a member of F(Ph). Besides that, we show the uniqueness of a minimal tree which belongs to F(P3) and characterize P3-(super)magic trees.