Interval Graphs are Reconstructible
Irene Heinrich, Masashi Kiyomi, Yota Otachi, Pascal Schweitzer
TL;DR
The paper resolves the reconstruction question for interval graphs of size at least three by introducing a novel trio of techniques: reconstruction by separation (via clean clique separators with linearly ordered neighborhoods), annotated induced subgraphs combined with distant-vertex information, and a resilient structure theory that partitions any interval graph into a bulk and up to two flanks (or outsiders when no flanks exist). By carefully reconstructing annotated subgraphs and exploiting the linear orderings, the authors show that the deck determines the graph up to isomorphism, and they further prove that this reconstruction can be performed in polynomial time. The key contributions are the Reconstruction-by-Separation lemma, the Distant Vertex Lemma, and a comprehensive decomposition into bulk, flanks, and outsiders that together overcome prior roadblocks in applying graph-structure theory to reconstruction. This advances both the theory of graph reconstruction and practical recognition/reconstruction of interval graphs, with potential extensions to bounded-treewidth graphs and related classes.
Abstract
A graph is reconstructible if it is determined up to isomorphism by the multiset of its proper induced subgraphs. The reconstruction conjecture postulates that every graph of order at least 3 is reconstructible. We show that interval graphs with at least three vertices are reconstructible. For this purpose we develop a technique to handle separations in the context of reconstruction. This resolves a major roadblock to using graph structure theory in the context of reconstruction. To apply our novel technique, we also develop a resilient combinatorial structure theory for interval graphs. A consequence of our result is that interval graphs can be reconstructed in polynomial time.