A Computational Search for Minimal Obstruction Graphs for the Lov\'{a}sz--Schrijver SDP Hierarchy

TL;DR

This paper introduces a modular framework for certifying membership in LS+ relaxations and uses it to computationally identify minimal graphs with specific LS+ ranks, revealing new structural insights.

Contribution

The paper presents LS+ certificate packages for transparent proofs and computationally finds new minimal graphs with higher counts, enhancing understanding of extremal graph structures.

Findings

Identified at least 49 non-isomorphic 3-minimal graphs.

Found at least 4,107 non-isomorphic 4-minimal graphs.

Determined smallest vertex-transitive graphs for ranks up to 4.

Abstract

We study the lift-and-project relaxations of the stable set polytope of graphs generated by $\text{LS}_+$, the SDP lift-and-project operator devised by Lov\'{a}sz and Schrijver. Our focus is on $\ell$-minimal graphs: graphs on $3\ell$ vertices with $\text{LS}_+$-rank $\ell$, i.e., the smallest graphs realizing rank $\ell$. This manuscript makes two complementary contributions. First, we introduce $\text{LS}_+$ certificate packages, a modular framework for certifying membership in $\text{LS}_+$-relaxations using only integer arithmetic and simple, concise calculations, thereby making numerical lower-bound proofs more transparent, reliable, and easier to verify. Second, we apply this framework to a computational search for extremal graphs. We prove that there are at least 49 non-isomorphic 3-minimal graphs and at least 4,107 non-isomorphic 4-minimal graphs, improving the previously known counts of 14 and 588, respectively. Beyond the increase in counts, the new examples sharpen the emerging structural picture: stretched cliques remain central but are not exhaustive, clique number is informative but not decisive, and some extremal graphs exhibit previously unseen graph minor and edge density behaviour. We also determine the smallest vertex-transitive graphs of $\text{LS}_+$-rank $\ell$ for every $\ell \leq 4$.