论文标题
通过图形überhomology对连接的统治进行分类
Categorifying connected domination via graph überhomology
论文作者
论文摘要
überHomology是一种最近定义的简单复合物的同源理论，可在图上产生微妙的信息。我们证明，大胆的同源性是überhomology的某种专业化，与图形中的主体集有关。为此，我们将überHomology解释为POSET同源性，并研究其功能性能。然后，我们表明图形的大胆同源性的欧拉特征与对其连接的统治多项式的评估相吻合。更重要的是，大胆的链复合物缩回由连接的主导集产生的复合物。我们在图形系列的几项同源性方面进行了几次计算。这些包括树木的消失结果，以及完整图的表征结果。
Überhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, a certain specialisation of überhomology, is related to dominating sets in graphs. To this end, we interpret überhomology as a poset homology, and investigate its functoriality properties. We then show that the Euler characteristic of the bold homology of a graph coincides with an evaluation of its connected domination polynomial. Even more, the bold chain complex retracts onto a complex generated by connected dominating sets. We conclude with several computations of this homology on families of graphs; these include a vanishing result for trees, and a characterisation result for complete graphs.
