Block structures of graphs and quantum isomorphism

TL;DR

This work analyzes how quantum isomorphism interacts with the block decomposition of graphs. By introducing anchored graphs and the Gamma/Delta framework, the authors inductively decompose connected graphs into simpler block structures while preserving qi isomorphisms, yielding that quantum isomorphic graphs have isomorphic block trees and block graphs with corresponding blocks qi isomorphic. Consequently, 2-connectedness is preserved under quantum isomorphism and quantum isomorphism respects both the number and arrangement of blocks and cut vertices, narrowing the search for minimal quantum-non-isomorphic pairs. The results also underpin algorithmic and algebraic studies of block graphs and quantum automorphism groups, with corollaries on necessary conditions for minimal counterexamples and implications for partitioned and anchored graph structures.

Abstract

We prove that for every pair of quantum isomorphic graphs, their block trees and their block graphs are isomorphic, and that such an isomorphism can be chosen so that the corresponding blocks are quantum isomorphic -- in particular, 2-connectedness is preserved under quantum isomorphism. We conclude with some corollaries, including obtaining some necessary conditions on a pair of quantum isomorphic, not isomorphic graphs with a minimal number of vertices.

Figures (2)

• Figure 1: Operation $\Gamma$ when the anchor consists of a cut vertex. The anchors are denoted by dashed lines.
• Figure 2: Operation $\Gamma$ when the anchor is a block. The anchors are denoted by dashed lines.

Theorems & Definitions (61)

• Proposition 1.1
• Proposition 1.2
• Theorem 1.3
• proof
• Theorem 1.4: Theorem 7.16 in Mancinska2019
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3