On the complexity of graph reconstruction

Abstract

In the wake of the resolution of the four color conjecture, the graph reconstruction conjecture has emerged as one focal point of graph theory. This paper considers the computational complexity of decisions problems (DECK CHECKING and LEGITIMATE DECK), the construction problems (PREIMAGE CONSTRUCTION), and counting problems (PREIMAGE COUNTING) related to the graph reconstruction conjecture. We show that: 1. DECK CHECKING ≤lm GRAPH ISOMORPHISM ≤lm LEGITIMATE DECK, and 2. if the graph reconstruction conjecture holds, then GRAPH ISOMORPHISM ≡liso DECK CHECKING. Eelatedly, we display the ftrst natural Gl-hard NP set lacking obvious padding functions. Finally, we show that LEGITIMATE DECK, PREIMAGE CONSTRUCTION, and PREIMAGE COUNTING are solvable in polynomial time on planar graphs, graphs with bounded genus, and partial k-trees for fixed k.