Given a finite undirected graph $X$, a vertex is $0$-dismantlable if its open neighbourhood is a cone and $X$ is $0$-dismantlable if it is reducible to a single vertex by successive deletions of $0$-dismantlable vertices. By an iterative process, a vertex is $(k+1)$-dismantlable if its open neighbourhood is $k$-dismantlable and a graph is $k$-dismantlable if it is reducible to a single vertex by successive deletions of $k$-dismantlable vertices. We introduce a graph family, the cubion graphs, in order to prove that $k$-dismantlabilities give a strict hierarchy in the class of graphs whose clique complex is non-evasive. We point out how these higher dismantlabilities are related to the derivability of graphs defined by Mazurkievicz and we get a new characterization of the class of closed graphs he defined. By generalising the notion of vertex transitivity, we consider the issue of higher dismantlabilities in link with the evasiveness conjecture.
NPA Hierarchy for Quantum Isomorphism and Homomorphism Indistinguishability
Technical University of Denmark（丹麦技术大学）；Instytut Matematyczny Polskiej Akademii Nauk（波兰科学院数学研究所）；QMATH, University of Copenhagen（哥本哈根大学 QMATH 中心）；IT-Universitetet i København（哥本哈根信息技术大学）；Department of Algebra, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院代数系）
A Computational Search for Minimal Obstruction Graphs for the Lovász--Schrijver SDP Hierarchy
Department of Mathematics and Statistics, University of Saskatchewan（萨斯喀彻温大学数学与统计系）；Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo（滑铁卢大学数学学院组合优化系）
A hierarchy of edge-weight symmetries in perfect matchings
MTA-ELTE Matroid Optimization Research Group and HUN-REN–ELTE Egerváry Research Group, Department of Operations Research, ELTE Eötvös Loránd University, and HUN-REN Alfréd Rényi Institute of Mathematics（MTA-ELTE拟阵优化研究组和HUN-REN–ELTE埃格瓦里研究组，ELTE罗兰大学运营研究系，以及HUN-REN阿尔弗雷德·雷尼数学研究所）；Department of Operations Research, ELTE Eötvös Loránd University（ELTE罗兰大学运营研究系）；Department of Information and Physical Sciences, Graduate School of Information Science and Technology, Osaka University（大阪大学情报理工学研究科信息物理科学系）
