Δ^P₂-complete lexicographically first maximal subgraph problems

Abstract

We prove that the lexicographically first maximal (lfm) connected subgraph problem for a graph property π is Δ^P₂-complete if π is hereditary, determined by the blocks, nontrivial on connected graphs, and polynomial-time testable. We also prove the Δ^P₂-completeness of the lfm induced path problem since the above result does not apply for this problem. Moreover, we analyze the lfm rooted tree problem for directed graphs. This problem can be shown to be Δ^P₂-complete but we show that it allows a polynomial-time algorithm when instances are restricted to topologically sorted directed acyclic graphs (dags). Further, the problem restricted to topologically sorted dags with degree at most 3 is shown in NC² while the problem for degree 4 is P-complete.