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 a 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: Relatedly, we display the first natural GI-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 e-trees for fixed k.

Complexity results in graph reconstruction

“…We strengthen the result of Mansfield [Man82] to show that, for every c ≥ 1, GI is polynomial-time isomorphic to LED c . For LVD c , we observe that for every c ≥ 1, GI ≤ p m LVD c (the c = 1 case of this already follows from a result of Kratsch and Hemaspaandra [KH94]). These results appear in Section 3.1.…”

“…The following results of Berman and Hartmanis [Har78,BH77a] give a sufficient condition for showing logspace or polynomial-time isomorphism between sets. (The wording of them used here mostly follows the presentation of [KH94].) We will use Theorem 2.8 to help us show isomorphism between GI and certain problems considered in this paper.…”

“…In this section, we investigate the complexity of VDC c , EDC c , LVD c , and LED c , for each c ≥ 1. Kratsch and Hemaspaandra [KH94] showed that GI is logspace isomorphic to VDC 1 .…”

“…Mansfield [Man82] and Kratsch and Hemaspaandra [KH94] studied complexity aspects of legitimate deck problems and deck checking problems. Mansfield [Man82] showed that LVD is polynomial-time many-one hard for the Graph Isomorphism problem (which we will often refer to as GI) and that LED is polynomial-time Turing equivalent to the Graph Isomorphism problem.…”

“…Mansfield [Man82] showed that LVD is polynomial-time many-one hard for the Graph Isomorphism problem (which we will often refer to as GI) and that LED is polynomial-time Turing equivalent to the Graph Isomorphism problem. Kratsch and Hemaspaandra [KH94] showed that LVD is logspace many-one hard for the Graph Isomorphism problem, proved that GI is logspace isomorphic to VDC, and obtained polynomial-time algorithms for LVD when restricted to certain classes of graphs-including graphs of bounded degree, partial k-trees for any fixed k, and graphs of bounded genus. Köbler, Schöning, and Torán [KST93] showed that if the Reconstruction Conjecture holds then LVD is in the complexity class LWPP.…”