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We give an example of two non isomorphic coordinate rings of a special kind of convex polyominoes whose Cohen–Macaulay types are generalized Fuss–Catalan numbers. We further provide a determinantal formula for these numbers.

In this work we deal with three variants of domination in graphs, these are Italian domination (or Roman {2}-domination), {2}-domination and 2-domination. We define Sicilian graphs as those graphs for which the Italian domination and the {2}-domination numbers coincide. Sicilian graphs constitute a superclass of Italian graphs (introduced by Klostermeyer and MacGillivray [J. Comb. Math. Comb. Comput. 108 (2019) 125–146]). First, we give a characterization of Italian graphs in terms of the existence of a special Roman {2}-dominating function. Then, we focus on web graphs for which their {2}-domination number was recently found (Cheng et al., Discrete Math. Algorithms App. 12 (2020) 2050055), and we study Sicilian web graphs. We explore also Sicilian co-bipartite graphs. As a by-product, we find the 2-domination number for web graphs and co-bipartite graphs. Finally, we show necessary conditions for non-Italian graphs to be Sicilian as well as characterize Sicilian graphs within some relevant graph classes such as quasi-threshold graphs and cographs.

Anchor technology is popularly employed in multi-view subspace clustering (MVSC) to reduce the complexity cost. However, due to the sampling operation being performed on each individual view independently and not considering the distribution of samples in all views, the produced anchors are usually slightly distinguishable, failing to characterize the whole data. Moreover, it is necessary to fuse multiple separated graphs into one, which leads to the final clustering performance heavily subject to the fusion algorithm adopted. What is worse, existing MVSC methods generate dense bipartite graphs, where each sample is associated with all anchor candidates. We argue that this dense-connected mechanism will fail to capture the essential local structures and degrade the discrimination of samples belonging to the respective near anchor clusters. To alleviate these issues, we devise a clustering framework named SL-CAUBG. Specifically, we do not utilize sampling strategy but optimize to generate the consensus anchors within all views so as to explore the information between different views. Based on the consensus anchors, we skip the fusion stage and directly construct the unified bipartite graph across views. Most importantly, $\ell _{1}$ norm and Laplacian-rank constraints employed on the unified bipartite graph make it capture both local and global structures simultaneously. $\ell _{1}$ norm helps eliminate the scatters between anchors and samples by constructing sparse links and guarantees our graph to be with clear anchor-sample affinity relationship. Laplacian-rank helps extract the global characteristics by measuring the connectivity of unified bipartite graph. To deal with the nondifferentiable objective function caused by $\ell _{1}$ norm, we adopt an iterative re-weighted method and the Newton’s method. To handle the nonconvex Laplacian-rank, we equivalently transform it as a convex trace constraint. We also devise a four-step alternate method with linear complexity to solve the resultant problem. Substantial experiments show the superiority of our SL-CAUBG.