论文标题
一个集团的盖子可以盖几个集团？
How many cliques can a clique cover cover?
论文作者
论文摘要
这项工作通过利用其集团的封面来检查图表上集团枚举的问题。应用包含/排除的原理用于确定集合$ \ MATHCAL {C} = \ {C_1，\ ldots，c_m \} $ of $ m $ cliques的$ \ Mathcal {c} = \ {c_1，\ ldots的尺寸$ r $的数量。这会更深入地检查所涉及的集合和轨道分区的$γ$，$ \ Mathcal {p}（\ Mathcal {n} _ {M} _ {M}）$ of $ \ MATHCAL {N} _ {N} _ {M} _ {m}该分区应用于集团，可深入了解集团枚举，并在集团封面中的集团中产生新的结果，包括大小$ r $的集团数量的表达式，以及这些图表上集团的功能。商图模型该分区提供了一个简洁的表示，以确定图形联合中的集团和最大集团。该分区还为相关问题提供了一个自然而强大的框架，例如通过相交集合与极端集合理论的联系来列举引起的连接组件。
This work examines the problem of clique enumeration on a graph by exploiting its clique covers. The principle of inclusion/exclusion is applied to determine the number of cliques of size $r$ in the graph union of a set $\mathcal{C} = \{c_1, \ldots, c_m\}$ of $m$ cliques. This leads to a deeper examination of the sets involved and to an orbit partition, $Γ$, of the power set $\mathcal{P}(\mathcal{N}_{m})$ of $\mathcal{N}_{m} = \{1, \ldots, m\}$. Applied to the cliques, this partition gives insight into clique enumeration and yields new results on cliques within a clique cover, including expressions for the number of cliques of size $r$ as well as generating functions for the cliques on these graphs. The quotient graph modulo this partition provides a succinct representation to determine cliques and maximal cliques in the graph union. The partition also provides a natural and powerful framework for related problems, such as the enumeration of induced connected components, by drawing upon a connection to extremal set theory through intersecting sets.
