On the computational difficulty of the terminal connection problem

A connection tree of a graph G for a terminal set W is a tree subgraph T of G such that leaves( T ) ⊆ W ⊆ V(T) . A non-terminal vertex is called linker if its degree in T is exactly 2, and it is called router if its degree in T is at least 3. The T erminal connection problem (TCP) asks whether G admits a connection tree for W with at most ℓ linkers and at most r routers, while the S teiner tree problem asks whether G admits a connection tree for W with at most k non-terminal vertices. We prove that, if r ≥ 1 is fixed, then TCP is polynomial-time solvable when restricted to split graphs. This result separates the complexity of TCP from the complexity of S teiner tree , which is known to be NP-complete on split graphs. Additionally, we prove that TCP is NP-complete on strongly chordal graphs, even if r ≥ 0 is fixed, whereas S teiner tree is known to be polynomial-time solvable. We also prove that, when parameterized by clique-width, TCP is W[1]-hard, whereas S Teiner tree is known to be in FPT. On the other hand, agreeing with the complexity of S teiner tree , we prove that TCP is linear-time solvable when restricted to cographs ( i.e. graphs of clique-width 2). Finally, we prove that, even if either ℓ ≥ 0 or r ≥ 0 is fixed, TCP remains NP-complete on graphs of maximum degree 3.