Efficient Isomorphism for $$S_d$$-Graphs and T-Graphs

An H-graph is one representable as the intersection graph of connected subgraphs a suitable subdivision fixed H, introduced by Biró et al. (Discrete Mathematics 100:267–279, 1992). proper if representing H can be chosen incomparable inclusion. In this paper, we focus on isomorphism problem for \(S_d\)-graphs and T-graphs, where \(S_d\) star with d rays T an arbitrary tree. Answering open Chaplick (2016, personal communication), provide FPT-time algorithm testing computing automorphism group when parameterized d, which involves classical group-computing machinery Furst (in Proceedings 11th southeastern conference combinatorics, theory, computing, congressum numerantium 3, 1980). We also show that at least hard posets bounded width, no efficient combinatorial-only known to date. Then extend our approach XP-time T-graphs size T. Lastly, contribute combinatorial in special case \(S_d\)- T-graphs.